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DYNAMIC METEOROLOGY 



By JORGEN HOLMBOE 
GEORGE E. FORSYTHE 
and WILLIAM GUSTIN 



DEPARTMENT OF METEOR- 
OLOGY, UNIVERSITY OF 
CALIFORNIA AT LOS ANGELES 



New York - JOHN WILEY AND SONS, INC. 
London - CHAPMAN AND HALL, LIMITED 



THIS BOOK HAS BEEN MANUFACTURED IN 
ACCORDANCE WITH THE RECOMMENDATIONS 
OF THE WAR PRODUCTION BOARD IN THE 
INTEREST OF THE CONSERVATION OF PAPER 
^JD OTHER IMPORTANT WAR MATERIALS. 



COPYRIGHT, 1945, BY 

JoRGBf MOLMBOE. WlLLIAM SHARP GUSTIN, 

AND GEORGE . FORSYTHE 



All Rights Reserved 

This book or any part thereof must not 
be reproduced in any form without 
the written permission of the publisher. 



PRINTED IN THE UNITED STATES OF AMERICA 



PREFACE 

This book is intended as a basic textbook in theoretical meteorology 
for students who are preparing for a professional career in meteorology. 
It may be helpful to students of such applied sciences as geophysics, 
aerodynamics, and hydrology, and to students of various branches of 
pure physics. 

Tfie aim of the book is to provide the theoretical background for the 
understanding of the physical behavior of the atmosphere and its mo- 
tions. Only material which is considered indispensable for the practical 
meteorologist and weather forecaster has been includ The book is 
self-contained and presupposes only some general knowledge of physics 
and calculus. Starting from the fundamental coiyepts of physics, it 
develops the thermodynamical and hydrodynamical principles by which 
atmospheric phenomena and the evolution of the weather may be 
explained. 

The theory of atmospheric motion is most naturally and conveniently 
developed in vector notation. The methods of vector algebra and some 
simple operations of vector calculus are therefore consistently used. No 
previous knowledge of vector methods is assumed ; the vector operations 
are introduced and explained gradually as the need arises as part of the 
general development of the subject. 

Since the book is intended as a basic introduction to the subject, few 
references to original papers are given. ',***' * T ' , 

The book originated from my lectures on dynamic meteorology given 
at the Massachusetts Institute of Technology from 1936 to 1940 and at 
the University of California at Los Angeles after 1940. In the selection 
and organization of the material I have been greatly aided by studies 
pursued under Professor V., Bjerknes at Oslo University in 1926-30, and 
under Professor C.-G. Rossby during my years at the Massachusetts 
Institute of Technology. In connection with the extensive training 
programs for weather officers for the armed forces my two co-authors, 
Messrs. Forsythe and Gustin, joined the department as instructors in 
dynamic meteorology. During the subsequent joint instruction of the 
course by the three authors the earlier mimeographed lecture material 
was revised and reorganized several times, and much new material was 
added, before the final version of the book was written. 

I am much indebted to Professor J. Bjerknes for his permission to 



PREFACE iv 

include in chapter 10 a large part of our joint paper, " On the Theory of 
Cyclones," Journal of Meteorology, Vol. 1, Nos. 1 and 2, 1944. I am 
also greatly obliged to Professor H. G. Hough ton of the Massachusetts 
Institute of Technology, whose unpublished lecture material on meteoro- 
logical thermodynamics was a great help when the first outline of 
chapters 2 and 3 was written. 

On behalf of the three authors I wish to extend sincere thanks to all 
those friends who have given us help and encouragement in our work. 
We are very grateful to Professor J. Kaplan, former chairman of the 
meteorological department at the University of California at Los 
Angeles, for his unfailing support during the preparation of the manu- 
script and illustrations. We also are much indebted to Professor H. U. 
Sverdrup and Professor M. Neiburger, who have read parts of the manu- 
script and made many helpful suggestions. 

JORGEN HOLMBOE 
UNIVERSITY OF CALIFORNIA 
AT Los ANGELES 
January 1945 



CONTENTS 

PAGES 

SYMBOLS AND CONSTANTS ix-xvi 

CHAPTER ONE. DIMENSIONS AND UNITS 

1-01 The goal of dynamic meteorology. 1-02 The tools of dynamic 
meteorology. 1*03 The fundamental variables and their dimensions. 
1'04 Mts units. 1-05 Comparison with cgs units. 1-06 Comparison 
with English units. 1-07 Pressure measurement and units 1-8 

CHAPTER Two. THERMODYNAMICS OF A PERFECT GAS 

2-01 Thermodynamical systems. 2-02 The physical variables. 2-03 Vol- 
ume. 2-04 Pressure. 2-05 Temperature. 2-06 Meteorological tem- 
perature scales. 2-07 Equation of state. 2-08 The perfect gas. 
2-09 Equation of state of a perfect gas. 2-10 Molecular weights. 
2-11 Universal gas constant. 2-12 Mixtures of perfect gases. 2-13 Mo- 
lecular weight of dry air. 2-14 Work in thermodynamics. 2-15 (,-/>)- 
diagram. 2-16 Isotherms of a perfect gas. 2-17 Heat. 2-18 The first 
law of thermodynamics. 2-19 Specific heats of gases. 2-20 Internal 
energy of a perfect gas. 2-21 Specific heats of a perfect gas. 2-22 Energy 
equations in logarithmic form. 2-23 Atmospheric processes. 2-24 Adia- 
batic processes of a perfect gas. 2-25 Adiabats on diagrams. 2-26 Poten- 
tial temperature. 2-27 Differentials and functions of state. 2-28 Entropy. 
2-29 Thermodynamic diagrams. 2-30 Important criteria of the diagram. 
2-31 Stiive diagram. 2-32 Emagram. 2-33 Tephigram 9-39 

CHAPTER THREE. THERMAL PROPERTIES OF WATER SUBSTANCE AND 

MOIST AIR 

3-01 Isotherms of water substance. 3-02 (a,e)-diagram and the triple 
state. 3-03 The critical state. 3-04 Thermal properties of ice. 
3-05 Thermal properties of water. 3-06 Equation of state of water vapor. 
3-07 Specific heats of water vapor. 3-08 Changes of phase. 3-09 Varia- 
tion of the latent heats with temperature. 3-10 Clapeyron's equation. 
3-11 Saturation vapor pressure over water. 3-12 Saturation vapor pres- 
sure over ice. 3-13 Pressure and temperature of melting. 3-14 Com- 
plete (T,e) -diagram. 3-15 Supercooled water. 3-16 Thermodynamic 
surface of water substance. 3-17 Moist air. 3-18 Moisture variables. 
3-19 Relative humidity. 3-20 Relations among the humidity variables. 
3-21 Numerical determination of mixing ratio. 3-22 Vapor lines on the 
diagrams. 3-23 Graphical determination of w, q, and e. 3*24 Thermal 
properties of moist air. 3-25 Equation of state of moist air; virtual tem- 
perature. 3-26 Specific heats of moist air. 3-27 Adiabatic process of 
unsaturated air. 3-28 Virtual potential temperature; characteristic point. 



CONTENTS vi 

3-29 Useful approximate formulas. 3-30 The adiabatic processes of sat- 
urated air. 3-31 Exact equation of the pseudo-adiabatic process. 
3-32 Exact equation of the reversible saturation-adiabatic process. 
3-33 Critique of the two equations. 3-34 Simplified equation of the adia- 
batic process of saturated air. 3-35 Isobaric warming and cooling. 
3-36 Graphical construction of the saturation adiabats. 3-37 Nomencla- 
ture. 3-38 Definitions of Wt O e , T aw , T ae , T d . 3-39 Definitions of T ie and 
Tiu,. 340 Surface observations. 341 Example 40-80 



CHAPTER FOUR. HYDROSTATIC EQUILIBRIUM 

4-01 The hydrostatic problem. 4-02 The fields of the physical variables. 
4-03 The coordinate system. 4-04 Analytical expression for scalar quanti- 
ties. 4-05 Vectors. 4-06 Vector sum. 4-07 Scalar product. 4-08 Me- 
chanical equilibrium. 4-09 The force of gravity. 4-10 The field of geo- 
potential. 4-11 Dynamic height. 4-12 Analytical expression for the 
force of gravity. 4-13 Potential vector; ascendent; gradient. 4-14 The 
pressure field in equilibrium. 4-15 The pressure gradient. 4-16 The 
hydrostatic equation. 4-17 Distribution of pressure and mass in equilib- 
rium. 4-18 The atmosphere with constant lapse rate. 4-19 The homo- 
geneous atmosphere. 4-20 The dry-adiabatic atmosphere. 4-21 The 
isothermal atmosphere. 4-22 Atmospheric soundings. 4-23 Graphical 
representation of dynamic height. 4-24 Adiabatic and isothermal layers. 
4-25 The Bjcrknes hydrostatic tables. 4-26 U. S. Weather Bureau hydro- 
static tables. 4-27 Height evaluation on a diagram. 4*28 Example of 
height computation on the diagram. 4-29 Further remarks. 4-30 U. S. 
standard atmosphere. 4-31 The pressure altimeter. 4-32 Altimeter 
errors 81-124 



CHAPTER FIVE. STABILITY OF HYDROSTATIC EQUILIBRIUM 

5-01 The parcel method. 5-02 Stability criteria. 5-03 The individual 
lapse rate. 5-04 The lapse rate in the environment. 5-05 Stability cri- 
teria for adiabatic processes. 5-06 Stable oscillation. 5-07 Finite 
displacement. 5-08 Latent instability. 5-09 The slice method. 
5-10 Formulas for 7 M and 7,. 5-11 Rate of precipitation. 5*12 Precipita- 
tion lines on the emagram 125-144 



CHAPTER Six. THE EQUATION OF MOTION 

6-01 Kinematics. 6-02 Velocity. 6-03 Differentiation of a vector. 
6-04 Acceleration. 6-05 Curvature. 6-06 Reference frames. 6-07 Dy- 
namics. 6-08 The force of gravitation. 6-09 The equation of absolute 
motion. 6-10 The acceleration of a point of the earth. 6-11 Zonal flow. 
6-12 Angular velocity. ^6-13 The vector product. 6-14 The scalar triple 
product. 6-15 The velocity of a point of the earth. 6-16 Absolute and 
relative velocity. 6-17 Absolute and relative acceleration. 6-18 Abso- 
lute and relative zonal flow. 6-19 The equation of relative motion 145-172 



CONTENTS 



CHAPTER SEVEN. HORIZONTAL FLOW 



7-01 Horizontal flow. 7-02 Natural coordinates for horizontal flow. 
7-03 Standard and natural components. 7-04 The acceleration. 
7-05 Cyclic sense. 7-06 Angular radius of curvature. 7-07 Horizontal 
curvature. 7-08 Geodesic curvature. 7-09 Vertical curvature. 7-10 The 
angular velocity of the earth. 7-11 The Coriolis force. 7-12 The pres- 
sure force and the force of gravity. 7-13 The component equations of 
relative motion. 7-14 The vertical equation. 7-15 The tangential 
equation. 7-16 The normal equation. 7-17 Geostrophic flow. 7-18 In- 
ertial flow. 7-19 Cyclostrophic flow. 7-20 Arbitrary horizontal flow. 
7-21 Maximum speed. 7-22 Solution of the normal equation. 7-23 Hori- 
zontal curvature of the streamlines 173-209 

CHAPTER EIGHT. WIND VARIATION ALONG THE VERTICAL 

8-01 Geostrophic gradient flow. 8-02 Isobaric slope. 8-03 The thermal 
wind equation. 8-04 Isothermal slope. 8-05 The approximate thermal 
wind equation. 8-06 Analysis of the shear hodograph. 8-07 Fronts. 
8-08 The dynamic boundary condition. 8-09 Application of the dynamic 
boundary condition. 8-10 The kinematic boundary condition. 8-11 Front 
separating two arbitrary currents. 8-12 The zonal front. 8-13 The 
geostrophic front 210-232 

CHAPTER NINE. WIND VARIATION ALONG THE VERTICAL 
IN THE SURFACE LAYER 

9-01 Dynamics of friction. 9-02 Steady great circle flow. 9-03 The 
viscous stress. 9-04 The viscosity of a perfect gas. 9-05 Viscosity of air 
and water. 9-06 The f notional force. 9-07 Total mass transport in the 
surface layer. 9-08 Wind distribution in the surface layer. 9-09 Rela- 
tion between surface velocity and geostrophic velocity. 9-10 The geo- 
strophic wind level. 9-11 The eddy viscosity 233-249 



CHAPTER TEN. MECHANISM OF PRESSURE CHANGES 

10-01 Equation of continuity. 10-02 Divergence. 10-03 Horizontal di- 
vergence. 10-04 Individual and local change. 10-05 Relative change in 
a moving pressure field. 10-06 The pressure tendency. 10-07 The tend- 
ency equation. 10-08 The aclvective pressure tendency. 10-09 Relation 
between the horizontal divergence and the field of pressure. 10-10 Longi- 
tudinal divergence in wave-shaped isobar patterns. 10-11 Critical speed 
in sinusoidal waves. 10-12 Transversal divergence in wave-shaped isobar 
patterns. 10-13 Total horizontal divergence associated with wave-shaped 
isobar patterns. 10-14 Barotropic waves in a westerly current. 10-15 The 
relative streamlines. 10-16 Stable baroclinic waves. 10-17 The first 
formation of the baroclinic wave. 10-18 Horizontal divergence in closed 
cyclonic isobar patterns. 10-19 Closed cyclonic isobar patterns sur- 
mounted by wave-shaped patterns. 10-20 Closed anticyclonic isobar 
patterns 250-294 



CONTENTS viii 

CHAPTER ELEVEN, CIRCULATION AND VORTICITY 

11-01 Method of line integrals. 11-02 Line integral of a vector. 
11-03 Line integrals of the equation of absolute motion. 11-04 Primitive 
circulation theorems in absolute motion. 11-05 The theorem of solenoids. 
11-06 Practical forms of the theorem of solenoids. 11-07 Dynamic bal- 
ance of steady zonal motion. 11-08 Thermal wind in zonal motion. 
11 09 Circulation theorems in developed form. 11-10 Transformation of 
the acceleration integral for closed individual curves. 11-11 Individual 
circulation in absolute motion. 11-12 Circulation of the latitude circles in 
zonal motion. 11-13 Relation between absolute and relative circula- 
tion. 11-14 Individual circulation relative to the earth. 11-15 Circula- 
tion of the circles in a local vortex. 11-16 Vorticity . 11-17 The vorticity 
in rectangular coordinates. 11-18 The vorticity in natural coordinates. 
11-19 Absolute and relative vorticity. 11-20 The theorem of absolute 
vorticity. 11-21 The theorem of relative vorticity. 11-22 Air current 
crossing the equator. 11-23 Air current crossing a mountain range. 
11-24 Non-diverging wave-shaped flow pattern. 11-25 Export. 
11-26 Irrotational vectors. 11-27 Velocity potential. 11-28 Stream 
function 295-336 

CHAPTER TWELVE. THEORY OF WAVES IN A ZONAL CURRENT 

12-01 The atmospheric equations. 12-02 Autobarotropy. 12-03 Bound- 
ary conditions. 12-04 Sinusoidal waves in a westerly current. 
12-05 Waves in a barotropic current. 12-06 The pressure field in the 
barotropic wave. 12-07 The speed of propagation of the barotropic wave. 
12-08 Waves in a baroclinic current 337-363 

INDEX 365-378 



SYMBOLS AND CONSTANTS 
(All numerical values are in mts mechanical units.) 

1. SCALARS 

a = 6.371 X 10 6 Mean radius of earth (6-08). 
as = 6.378 x 10 6 Equatorial radius of earth (6-08). 
a P = 6.357 X 10 6 Polar radius of earth (6-08). 
1 atm = 101.33 Normal atmosphere (1-07). 

A Area. 

A, AS, A R Amplitude of wave-shaped path (11-24), 
streamline (10-11), relative streamline 
(10-15). 
A so Amplitude factor (12-05). 

b n Horizontal normal component of pressure 
force (7-16). 

c Curve; closed curve. 
c Specific heat (2-17). 
c Average speed of heat motion (9-04). 
c Speed of propagation (10-14); see also c. 
CH Horizontal path (7-07). 

c p , c v Specific heat at constant pressure, constant 
volume: of perfect gas (2-19); of moist 
air (3-26). 
c p d = 1004 Specific heat of dry air at constant pressure 

(2-21). 
Cvd - 717 Specific heat of dry air at constant volume 

(2;21). 
CPV= 1911 Specific heat of water vapor at constant 

pressure (3-07). 
c w = 1450 Specific heat of water vapor at constant 

volume (3-07). 

c w = 4185 Specific heat of water (3-05). 
d = 2060 Specific heat of ice (3-04). 
cb Centibar (1-04). 
C Heat capacity (2-17). 



SYMBOLS AND CONSTANTS 



C Circulation: general (11-02); relative to 

earth (11-13). 
C a Absolute circulation (11-11). 

d Individual (process) differentiation symbol 

(4-10, 10-04). 
dyn Dynamic (4-11). 
D Displacement in 12 hours (7-17). 
d Partial (local) differentiation symbol 
(2-20, 10-04). 

e 2.71828-.- . 
e Vapor pressure (3-01, 3-18). 
e 8 Saturation vapor pressure (3-08, 3-19) ; satu- 
ration pressure between any two phases 
(3-10). 
E Export (11-25). 

F Transport capacity (10-09, 10-10). 

g Acceleration of gravity (4-09) ; see also g. 
g a Acceleration of gravitation (6-08). 
A, g* Apparent gravity (8-12); virtual gravity 

(7-14). _ 
g#=9.78 Acceleration of gravity at the equator 

(4-09). 

g p = 9.83 Acceleration of gravity at the poles (4-09). 
45 = 9.80617 Acceleration of gravity at 45 latitude 

(4-09). 
g n = 9.80665 Standard value of acceleration of gravity 

(1-07). 
Gravitational constant (6-08). 

h Heat per unit mass (2-17). 

> h p Thickness : of geopotential unit layer (4- 10) ; 

of isobaric unit layer (4-lo). 

// Heat (2-17). 

H Height in dynamic meters. 

77 Subscript denoting horizontal component. 

H n Distance between 5-millibar isobars (7-17). 

H T Distance between 5-degree isotherms (8-05). 

k Wave number (10-11). 
k Constant used in computing dynamic height 
(4-27). 



G = 6.658 x 10 



1-8 



xi SYMBOLS AND CONSTANTS 

kj Kilojoule (1-04). 

K Curvature (6-05); horizontal curvature of 

path (10-10). 

Kg Geodesic curvature (7-08). 

KH Horizontal curvature of path (7-07). 

KHS, KS Horizontal curvature of streamline 
(7-23, 10-10). 

/ Arbitrary linear coordinate or line (4*05 ). 

/ Subscript denoting component along 1. 

/ Mixing length (9-11). 

In Natural logarithm (2-22). 

log Logarithm to base 10. 

L Mean free path (9-04). 

L, LS Wave length: of path (11-24); of stream- 
line (10-11). 

[L] Dimension of length (1-03). 

L = 2.500 x 10 6 Latent heat of evaporation (3-08). 

L iv = 2.834 X 10 6 Latent heat of sublimation (3-08). 

Liw = 0.334 X 10 6 Latent heat of melting (3-08). 

Li2 Any one of L, L iv , L iw (3-08). 

m Molecular weight (2-10). 

m Mass of molecule (9-04). 

m d = 28.97 Molecular weight of dry air (2-13). 

m v = 18.016 Molecular weight of water vapor (3*06). 

m Meter (1-04). 

mts Meter- ton-second (1-04). 

[M] Dimension of mass (1-03). 

M Mass. 

M - 5.988 X 10 21 Mass of the earth (6-08). 

n Horizontal coordinate normal to path, in- 
creasing to left (7-02). 

n Subscript denoting component along n. 

n Tt / 7 (5-10). 

n Angular wave number (10-11). 

N Number of solenoids (11-05). 

N Number of molecules per unit volume 
(9-04). 

Origin (6-01). 

p Pressure (2-04). 

pd Partial pressure of dry air (3-20). 



SYMBOLS AND CONSTANTS xii 

p s Pressure at characteristic point (3-28). 

P Point. 

P Procession (11-02). 

P Rate of precipitation (S 1 1 ) . 

q Specific humidity (3-18). 

q 8 Saturation specific humidity (3-19). 

r Length of position vector r (6-01). 

r Distance from center of earth (4-09, 6-08). 

r Relative huimdity (3-19). 

rad Radian (1-04). 

R Radius of curvature (6-05). 

RH Radius of horizontal curvature (7-07). 

RS Radius of horizontal curvature of stream- 
line (11-18). 

R Specific gas constant: of peffect gas (2-09); 

of moist air (3-25). 

R* = 8313.6 Universal gas constant (2-11). 

R d = 287.0 Specific gas constant for dry air (2-09). 

R v = 461.5 Specific gas constant for water vapor (3-06). 

s Arc length (6*02); horizontal coordinate 

along path (7-02). 

5 Specific entropy (2-28). 

5 Subscript denoting component along t. 

- s Second of time (1-04). 

5, n, z Natural coordinates (7-02). 

/ Time. 

/ Temperature in degrees centigrade (2-05). 

t (Metric) ton (1-04). 

T Temperature in degrees absolute (2-05). 

[T] Dimension of time (1-03). 

r* Virtual temperature (3-25). 

r a , Td, T aw > Ti W , Tie, Tae Temperature parameters of moist air 

(3-28, 3-38, 3-39). 

T p Standard temperature (4-30). 

T p Temperature on isobaric surface (8'03). 

u Magnitude of geostrophic deviation velocity 

(9-08). 

u Specific internal energy (2-18). 

U Internal energy (2-18), 



xiii SYMBOLS AND CONSTANTS 

v Molar volume (2-10). 

VQ - 22,414 Molar volume at 1 atm, C (2-10). 

v Speed, magnitude of velocity (6-02). 

v e Absolute speed of a point of the earth (6-1S). 

v Geostrophic speed (7-17). 

Vi Inertial speed (7-18). 

v c Cyclostrophic speed (7-19). 

VQ, Vi* VG Speed parameters in horizontal flow (7-22). 

v c Critical speed (10-10). 

v Mean zonal wind speed (10-10). 

Ay Parameter of wave-shaped flow pattern 

(10-10). 

V Volume. 

w Specific work (2-14). 

w Mixing ratio (3-18). 

w 8 Saturation mixing ratio (3-19). 

W Work (2-14). 

x, y, z Rectangular (Cartesian) coordinates; stand- 
ard coordinates with x axis toward the 
east, y axis toward the north, z axis 
toward the zenith (4-03). 
x, y, z Subscripts denoting components along x, y, 

z axes. 

z p Standard altitude (4-30). 
z p Altitude of isobaric surface (8-02). 

a. Specific volume (2-03). 

o = 773 Specific volume of dry air at 1 atm, C 
(2-09). 

7 Lapse rate of temperature. 

7* Lapse rate of virtual temperature (418). 

y d = 0.000996 Dry-adiabatic lapse rate (4-20). 

yn = 0.00348 Lapse rate in homogeneous atmosphere 

(4-19). 

7, Saturation-adiabatic lapse rate (5*03). 

7 Unsaturated-adiabatic lapse rate (5-03). 

5 Geometric differentiation symbol (4-10). 
e = 0.622 m v /m d (3-06), 



SYMBOLS AND CONSTANTS xiv 



f Vertical component of vorticity: general 

(11-16); relative to earth (11-19). 
f a Vertical component of absolute vorticity 
(11-19). 

rj Acoustical constant c p /c v : for perfect gas 

(2-24); for moist air (3-27). 
= 1.400 Acoustical constant for dry air (2-24). 

Angle. 

6 Angular radius of curvature (7-06). 
Potential temperature (2-26, 3-27). 
O w , 6 e Potential temperature parameters of moist 

air (3-38). 

p OT, Of Angle of inclination between level surface 
and: isobaric surface (8-02); isothermal 
surface (8-04); frontal surface (8-09). 
[0] Dimension of temperature (1-03). 

K Poisson's constant R/c p : for perfect gas 

(2-24); for moist air (3-27). 
= 0.286 Poisson's constant for dry air (2-24). 
K Roughness parameter (9-09). 

\s Angular wave length of streamline (10-11). 
/i, ju c Viscosity (9-03); eddy viscosity (9-10). 
v Circular frequency (5-06). 

TT 3.14159 

TT Barotropic pressure function (12-06). 
TT, TH Osculating plane (7-07); horizontal plane 
(7-07). 

p Density (2-03). 

GS, a p Angular amplitude of wave-shaped stream- 
line (10-11), isobar (10-10). 
S Equatorial projection of area enclosed by 
closed fluid curve (11-13). 

r, TJ, r p Period (5.-06); inertial period (7-18); pen- 
dulum day (7-18). 



xv SYMBOLS AND CONSTANTS 

tp Thcrmodynamic potential (3-10). 

V Velocity potential (11-27). 

V Latitude (7-10). 

<j> Geopotential ; dynamic height in dynamic 

decimeters (4-10). 

<t>d, </>A, <t>i Dynamic height of: dry-adiabatic atmos- 
phere (4-20); homogeneous atmosphere 
(4-19); atmosphere with linear lapse rate 
(4-18). 
<t> a Gravitational potential (6-08). 

fa Centrifugal potential (6-10). 

\t/ Angle indicating orientation of tangent to a 

curve (6-05). 
\l/ Stream function (11-28). 

o) Angular speed: general (6-05); relative to 

earth (6-11). 

c* a Absolute angular speed (6-11). 

fl = 7.292 X 10~ 5 Angular speed of the earth (6-10, 7-10). 
8* | Q| sin? (7-10). 

2. VECTORS 

| a | Magnitude of vector a (4-05). 

a'b Scalar product (4-07). 

axb Vector product (6-13). 

A Vector area (10-08). 

b Pressure force per unit mass (6-09). 

C Coriolis force (6-19). 

c Velocity of propagation (10-05). 

V Del (4-12, 10-01). 

Vs Ascendent of scalar e (4-13). 

V//s Horizontal ascendent of scalar s (4- IS). 

Va Divergence of vector a (10-01). 

V#*a Horizontal divergence of vector a (10-03). 

f a Inertial force (6-09). 
i e Centrifugal force of the earth (6-10). 
F Mass transport (9-07). 
F Vector perpendicular to frontal surface 
(8-08). 



SYMBOLS AND CONSTANTS xvi 



g Force of gravity (4-09, 6-10). 
g a Force of gravitation (4-09, 6-08). 

i, j, k Rectangular (Cartesian) system of unit 
vectors along x, y, z axes (4-06). 

K Vector curvature (6-05). 
1 Unit vector along arbitrary line (4-05). 
m Frictional force per unit mass (9-01). 

n Horizontal unit normal, pointing to left of 

path (7-02). 
N Unit normal toward center of curvature 

(6-05). 

n/r, n^r Horizontal unit vector normal to: front 
(8-09); isotherm (8-04). 

Zero-vector (4-06). 

r Position vector (6-01). 
R Vector radius of curvature (4-09, 6-05). 

t Unit tangent vector (6-02). 

t, n, k Natural system of unit vectors along 5, n, z 
directions (7-02). 

u Geostrophic deviation velocity (9-02). 

v Velocity: general (6-02); relative to earth 

(6-16). 

v Absolute velocity (6-16). 
v e Velocity of a point of the earth (6-15). 
V0 Geostrophic velocity (8-01). 
v Acceleration (6-05); acceleration relative tc 
earth (6-17). 

T Viscous stress (9-03, 9-04). 

co Angular velocity (6-12). 
Q, Angular velocity of the earth (6-12). 
Q z Local angular velocity (7-11). 



CHAPTER ONE 
DIMENSIONS AND UNITS 

1-01. The goal of dynamic meteorology. It is customary to divide 
meteorology into several fields, of which dynamic meteorology is one. 
Dynamic meteorology starts from pure physical theory and attempts to 
give a systematic and quantitative description of the composition and 
physical behavior of the atmosphere. The goal is the complete explana- 
tion in physical terms of the atmospheric phenomena constituting the 
weather. Synoptic meteorology (another of the fields) starts from 
weather observations and attempts to describe the current state of the 
weather in such terms that its future development may be predicted. 
The goal is to forecast the weather without error. 

It is clear that the ultimate goals of dynamic and synoptic meteorology 
can only be attained simultaneously: the first perfect forecaster would be 
the first man who could explain completely the physical behavior of the 
atmosphere, and vice versa. 

In the present stage of meteorological development certain steps have 
been taken in the direction of these goals. It must be understood that 
these steps are only a beginning; for the most part the goals still remain 
unattained. Nevertheless it is already clear that an understanding of 
the atmosphere in physical terms is absolutely essential for the synoptic 
meteorologist. 

1-02. The tools of dynamic meteorology. Knowing the goal of 
dynamic meteorology, we first select the branches of physics which fur- 
nish suitable tools. These seem at present to be thermodynamics and 
hydrodynamics. Accordingly, chapters 2 and 3 are devoted to thermo- 
dynamics, and chapters 4 and 6 to hydrodynamics. The other chapters 
are/devoted to more properly meteorological topics. 

The presentations will presuppose a certain knowledge of general 
physics and of the calculus. We will start from elementary physical 
principles and build up all the thermodynamics, vector analysis, and 
hydrodynamics used in this book. From the beginning the notation 
and subject matter will be adapted exclusively to the needs of meteorol- 
ogy. This physical material constitutes a background indispensable for 
the understanding of even the most elementary atmospheric phenomena. 

In chapter 1 are introduced some important mechanical variables of 
general physics, together with their dimensions and units. 

1 



Section 1-03 2 

1-03. The fundamental variables and their dimensions. A mechani- 
cal system is measured by various quantities, such as force and energy. 
All these quantities are reducible to three fundamental quantities, 
namely, mass, lengthy and time. All other mechanical quantities can 
be expressed in terms of these three fundamental quantities and are 
called derived quantities. (It should be understood that the selection of 
fundamental quantities is by no means unique, but our choice has the 
advantage of simplicity.) 

The method by which the derived quantities are built up from the 
fundamental quantities is best expressed in terms of algebraic expres- 
sions called dimensions. There is assigned to each of the fundamental 
quantities a dimensional letter in brackets, as follows: 

(1) [mass]- [M]; 

(2) [length] -[L]; 

(3) [time] - [T]. 

For completeness, we include a fourth fundamental quantity which we 
will need in thermodynamics, namely, temperature: 

(4) [temperature] = [0]. 

A pure number, for example, an angle expressed in radians or a molecular 
weight, is assigned the dimension unity: 

(5) [pure number] - [1] - [LMT]. 

Since M, L, and T not enclosed in brackets will have other meanings in 
this book, it is essential that dimensional formulas always be enclosed in 
brackets. 

The derived quantities are assigned dimensions which are algebraic 
monomials in M, L, T, and 6. The exponents represent the powers of 
fundamental quantities contained in the derived quantity. It is 
assumed that the reader is familiar with the derived quantities of ele- 
mentary mechanics, but for convenience we give brief definitions and the 
dimensions of those .of especial use in dynamic meteorology. Many 
of these will also be discussed later; those involving temperature will be 
introduced in chapters 2 and 3. 

Area is ultimately reduced to the area of a rectangle, which is the 
product of the lengths of its sides: 

(6) [area] = [length] x [length] - [L 2 ]. 

Volume is ultimately reduced to the volume of a rectangular prism, 
which is the product of the base area and an altitude : 

(7) [volume] - [area] x [length] - [L 3 ]. 



3 Section 1-03 

Density is defined as the mass of an object per unit volume occupied 
by the object: 

(8) [density] = [mass] + [volume] = [ML"" 3 ]. 

Specific volume is defined as the volume occupied by an object per unit 
mass of the object (" specific " always stands for " per unit mass "): 

(9) [specific volume] = [volume] * [mass] = [M~ 1 L 3 ]. 
Velocity is the distance traversed per unit of time: 

(10) [velocity] = [length] ^ [time] - [LT^ 1 ]. 
Acceleration is the change of velocity per unit of time : 

(11) [acceleration] = [velocity] + [time] = [LT~ 2 ]. 

Force is sometimes taken as a fundamental quantity, but is always 
found to be proportional to the mass of an object multiplied by the 
acceleration of the object produced by the force: 

(12) [force] = [mass] x [acceleration] = [MLT~ 2 ]. 

Pressure is defined to be the force exerted on a surface per unit area of 
surface: 



(13) [pressure] = [force] + [area] = 

The work done by a force is the product of the force and the length 
through which the force moves something: 

(14) [work] - [force] x [length] = [ML 2 T- 2 ]. 

Energy is the measure of the work which can be gotten out of a system 
by some procedure. It then has the same dimensions as work: 

(15) [energy] = [ML 2 T~ 2 ]. 
Specific work is the work done per unit mass: 

(16) [specific work] - [L 2 T~ 2 ]. 
Specific energy is the energy per unit mass: 

(17) [specific energy] - [L 2 T~ 2 ]. 

Angular velocity is the angular distance traveled per unit of time. 
Since [angle] = [1], we have: 

(18) [angular velocity] = [angle] 4- [time] [T""" 1 ]. 
Momentum is the mass of something times its velocity: 

(19) [momentum] - [mass] x [velocity] - [MLT"" 1 ]. 



Section 1-03 4 

Other quantities will be introduced later as the need for them arises. 
It should be remarked that a quantity expressed as a vector will be 
assigned the same dimensions as the same quantity expressed as a scalar. 

Any physical equation can be interpreted as a relation between dimen- 
sional quantities. The dimensions of the variables in an equation must 
satisfy the algebraic relation expressed by the equation. For example, 
the relation force = change of momentum per unit time can be expressed 
dimensionally by 

[MLT~ 2 ] - [MLT- 1 ] -i- [T]. 

The habit of checking the dimensions of every equation should be 
developed. If the dimensions do not balance, the formula is definitely 
wrong. If the dimensions do balance, the formula is probably correct 
up to a numerical constant or other quantity of dimension unity. In 
certain fields, dimensional analysis is used even to derive physical rela- 
tions and is a tool of great power. 

1-04. Mts units. The definitions of the physical quantities of 1-03 
are independent of the particular choice of units used to measure them. 
Their dimensions are also independent of the system of units; that is the 
peculiar advantage of dimensional analysis. 

But as soon as we desire to measure and assign numerical values to 
those quantities of 1-03, we must have a system of units. The usual 
procedure is first to define the fundamental units the units of length, 
mass, and time. Each derived unit is then defined by compounding its 
component fundamental units according to the dimensional definition of 
the derived unit. 

The standard centimeter-gram-second or cgs system of units is uni- 
versal and convenient in experimental physics, where the systems gener- 
ally considered are of the same order of magnitude as the units. In 
meteorology the system under consideration is the atmosphere, whose 
magnitude is enormous compared with the cgs units. Hence it is logical 
to introduce units which are of atmospheric magnitude, and which at the 
same time are easily translatable into the comparable cgs units. The 
system adopted by the International Meteorological Conference in 1911 
is the meter-ton-second or mts system, and we shall use it exclusively in 
this book. The units are defined as follows: 

a. The unit of length is one meter, abbreviated 1 m. This was 
intended to be one ten-millionth of the length of the meridian from the 
pole to the equator at sea level. Since the meridian is divided into 
90 degrees of latitude, each degree of latitude was to be a length unit 
equal to X 10 6 m, or 1 1 1 km. Although the original computation had 



5 Section 1-05 

a small error, the relation 
(1) 1 degree of latitude = 111.1 km 

is correct and often used in synoptic work. The meter is now fixed by a 
standard in Paris, but may always be reproduced in terms of the wave 
length of a certain line in the spectrum of cadmium. 

b. The unit of mass is one (metric) ton, abbreviated 1 1. It is the mass 
of one cubic meter of pure water at its maximum density (near 4C). 

c. The unit of time is one (solar) second, defined as I/ (24 60 60) of 
the mean time interval between consecutive upper transits of the sun 
across the same meridian. 

The derived units are obtained as follows: The units of area and 
volume are one square meter (1m 2 ) and one cubic meter (1 m 3 ) respec- 
tively. The units of density and specific volume are one ton per cubic 
meter (1 tm~ 3 ) and one cubic meter per ton (1 m 3 f 1 ), respectively. 
The units of velocity and acceleration are one meter per second (1m s"" 1 ) 
and one meter-per-second-per-second (1 m s~ 2 ), respectively. 

The unit of force is the force which gives a mass of one ton an accelera- 
tion of 1 m s~ 2 . Unfortunately it has no more specific name than one 
ton-meter-per-second-per-second (1 t m s~ 2 ) or one mts unit of force. 
The unit of pressure is the pressure developed by an mts unit of force 
acting on each square meter and is called one centibar (1 cb). The unit 
of work is the work done by one mts unit of force acting through a dis- 
tance of one meter, and is called one kilojoule (1 kj). The kilojoule 
serves also as the unit of energy. The unit of specific work and specific 
energy is one kilojoule per ton (1 kj t" 1 ). The origin of the names 
centibar and kilojoule will be explained in 1-05. 

Angles will be expressed in radians (rad), where 2ir radians equals 
360. The unit of angular velocity is one radian per second (1 rad s" 1 ). 

1-05. Comparison with cgs units. It is presumed that the reader is 
familiar with the cgs units, whose definitions are completely analogous 
to the mts units. They start from the fundamental units: 

(1) 1 centimeter (1 cm) = 10~ 2 meter; 

(2) 1 gram (1 gm) = 10~ 6 ton; 

(3) 1 second (Is) =1 second. 

The cgs unit of force, in contrast to the corresponding mts unit, has a 
name the dyne. The cgs unit of pressure is the barye. The cgs unit 
of work or energy is the erg. Because of the small size of the barye and 
erg, the following alternate measures are often introduced, but they are 



Section 1-05 6 

not strictly speaking units in the cgs system: 

pressure: 1 bar = 10 6 baryes; 

work or energy: 1 joule = 10 7 ergs. 

It is from these names that the mts units get their names. It will be 
shown presently that in accordance with their names: 

(4) 1 centibar = Iff" 2 bar; 

(5) 1 kilojoule = 10 3 joules. 

Despite their prefixes, it must be understood that the centibar and kilo- 
joule are actually the units of pressure and work in the mts system. 

In order to make a convenient reference page, we will tabulate (table 
1-05) the quantities so far considered. For each quantity, we give: 
(i) its name, (ii) its dimensions, (iii) its mts unit, (iv) its cgs unit, (v) the 
number N of cgs units contained in one mts unit of that quantity. For 
all the derived quantities used in this book the mts unit is equal to or 
larger in magnitude than the corresponding cgs unit; so that in table 1-05, 



TABLE 1-05 
(Mts unit = cgs unit X N) 



QUANTITY 


DIMENSIONS 


MTS UNIT 


Cos UNIT N 


Length 


[L] 


I m 


cm 10 2 


Mass 


[M] 


t 


gm 10 6 


Time 


[T] 


s 


s 1 


Area 


[L 2 ] 


m 2 


cm 2 10 4 


Volume 


[L 3 ] 


m 3 


cm 3 10 6 


Density 


[ML- 3 ] 


tm" 3 


gm cm~~ 8 1 


Specific volume 


[M^L 3 ] 


m 3 r 1 l 


cm 3 gm"" 1 1 


Velocity 


[LT- 1 ] 


m s" 1 1 


cm s" 1 10 2 


Acceleration 


[LT- 2 ] 


m s~ 2 1 


cm s~ 2 10 2 


Force 


[MLT~ 2 ] ] 


1 t m s~ 2 1 


gm cm s" 2 = 1 dyne 10 8 


Pressure 


[ML- 1 '?- 2 ] ] 


L t m 1 s~ 2 = 1 cb 1 


gm cm" 1 s"" 2 = 1 barye 10 4 


Work 1 
Energy / 


[ML 2 T~ 2 ] 


1 1 m 2 s~ 2 = 1 kj 1 


gmcm 2 s" 2 = lerg 10 lc 


Specific work 1 
Specific energy J 


[L 2 T~ 2 ] 


1 kj t- 1 1 


erg gm"" 1 10 4 


Angular velocity 


[T- 1 ] 1 


racl s" 1 1 


rad s" 1 1 


Momentum 


[MLT- 1 ] 1 


t m S" 1 1 


gm cm s" 1 10 8 



To obtain N readily, we start from the relations (1), (2), and (3). 
Thus for length [L], N =* 10 2 ; for mass [M], N - 10 6 ; and for time [T], 
N - 1. To obtain N for any derived quantity, we multiply together the 
component N's according to the dimensional formula. For example, 



7 Section 1-07 

pressure - \MLT 1 T*\. Hence, for pressure, N - 10 6 (lO 2 )" 1 (I)"" 2 - 
10 4 . This means that 1 cb = 10 4 baryes (= 10~ 2 bar), which proves 
(4). For work, N = 10 6 (10 2 ) 2 (l)" a = 10 10 . Thus 1 kj - 10 10 ergs 
(= 10 3 joules), which proves (5). 

It should be noted that density and specific volume have the same 
numerical values in both cgs and mts units, and that water has unit 
density and unit specific volume. 

1 -06. Comparison with English units. Even in countries where the 
metric system is not in general use, the weather services use metric units 
to a very large degree. Hence the reader should become familiar with 
the units of table 1-05 to the point where he can use them in everyday 
life. To help in this, we give a few comparisons with English units: 

Length: 1 m= 39.37 in.; 

10,000 ft = 3048m; 

1 km = 0.6214 mile f mile; 

Mass: It- 2205 (Ib mass) ; 

Density: 1 t m~ 3 = 62.43 (Ib mass)' ft"" 3 ; 

Velocity: 1 m s""" 1 = 2.237 miles hr" 1 2| miles hr"" 1 ; 

Pressure: 1 cb = 0.1450 (Ib force) in.- 2 | (Ib force) in." 2 ; 

Work: 1 kj = 737.6 ft-(lb force). 

1-07. Pressure measurement and units. Besides the centibar the 
meteorologist must know several other pressure units. In the weather 
services pressure is represented in millibars (mb) : 

(1) 1 cb - 10 mb. 

For this reason it is necessary to caution students repeatedly to convert 
pressure to centibars when computing with mts units. 

The standard pressure-measuring instrument of meteorology is the 
mercury barometer. This instrument is so designed that the pressure 
of the air is balanced against a column of mercury (Hg) whose length can 
be measured very accurately. The pressure is expressed as the length 
of the mercury column, in either millimeters or inches. The pressure 
of the mercury column is determined by its weight per unit cross sec- 
tion, which for a column of constant height varies with both the density 
and the acceleration of gravity. To compare pressure readings made 
with mercury barometers at different temperatures and at points where 
gravity is different, all readings are reduced to a standard tempera- 
ture and a standard value of gravity. For millimeter barometers, the 



Section 1-07 8 

standard temperature is 0C, at which the density p of mercury is 
13.5955 t m~ 3 . The standard value of the acceleration of gravity is 
g n = 9.80665 m s~ 2 , which is approximately the sea-level value of 
gravity at 45 latitude. 

The pressure of one normal atmosphere (1 atm) is that balanced by a 
column of mercury 760 mm = 0.760 m long, under the above standard 
conditions. It is the reference pressure of physical chemistry. To 
evaluate it, we find that 

1 atm = (0.76 m) x (13.5955 t m~ 3 ) x (9.80665 m s~ 2 ) = 101.33 cb. 
Thus the normal atmosphere is expressed in mechanical units. We have 

(1) 1 atm = 760 mm Hg = 29.92 in. Hg = 101.33 cb = 1013.3 mb. 

Another reference pressure frequently used in meteorology is standard 
pressure, which is defined to be exactly 100 cb. Thus 

(2) 750.04 mm Hg = 29.53 in. Hg = 100.00 cb = 1000.0 mb. 

From either (1) or (2), tables are computed to convert pressure from 
inches of mercury or millimeters of mercury to millibars. The conver- 
sion factor 

(3) 1 mb = f mm Hg 

is easy to remember and yields all the accuracy usually required for con- 
verting millibars into millimeters of mercury. 



CHAPTER TWO 
THERMODYNAMICS OF A PERFECT GAS 

2-01. Thennodynamical systems. Thermodynamics deals with 
systems which, in addition to certain mechanical parameters to be 
mentioned later, require for their description a thermal parameter, the 
temperature. The very definition of temperature requires that a system 
be in equilibrium. Thus of necessity thermodynamics is the study of 
systems in equilibrium and of processes which can take place in states 
differing only slightly from the state of equilibrium. The fact is that 
the actual atmosphere is not in equilibrium. Dynamic meteorology is 
compelled to make the pretense that equilibrium exists, in order to make 
an analysis. We should therefore expect the results to have some slight 
disagreements with conditions in the real atmosphere. 

The systems considered mostly in dynamic meteorology are infinitesi- 
mal parcels of: (i) dry air, which can for practical purposes be con- 
sidered as one substance; (ii) pure water substance in any one, two, or 
three of the phases solid (ice), liquid (water), or gas (water vapor); 
(iii) a mixture of dry air with some water vapor, called moist air; (iv) a 
mixture of moist air with some water droplets or ice crystals. 

2 '02. The physical variables. The infinitesimal systems considered 
will be described thermodynamically by the four parameters mass (dM), 
volume (8V), pressure (p), and temperature (T). One system will be 
defined so as always to consist of the same particles. Hence its mass 
dM and composition will remain constant, and the other parameters, 
volume, pressure, and temperature, will be called the physical variables. 
The values of the physical variables will completely describe the state 
of the system. 

203. Volume. The actual volume 8V is conveniently replaced by 
the specific volume a = 8V/8M. See 1-03(9). Since the mass dM 
remains constant, is directly proportional to the actual volume 8V. 
An alternative mass- volume variable is the density p = dM/dV. See 
1-03(8). The two variables are related by the equation 

(1) ap-1. 

By means of (1), one of the variables may be replaced by the other in any 
physical equation. Either of them may with equal right be taken as the 
mass variable in dynamics, but we shall usually prefer a, 

9 



Section 2-04 10 

2-04. Pressure. To define the pressure p of a thermodynamic sys- 
tem, we must first consider any fixed point P in the system, and any 
fixed direction / at P. We assume it possible to place a testing surface 
of small plane area dA at the point P, and orient the testing surface nor- 
mal to the direction /. The molecules of that part of the system on one 
side of the testing surface will bombard the area 6A, giving rise to a force 
dFin the direction /. (The molecules on the other side of SA must be dis- 
regarded in computing 8F.) 

Experiment shows that dF is proportional to 5A for a range of areas 8A 
which are neither so large as to exceed the size of the system, nor so small 
as to be of molecular dimensions. The proportionality factor is called 
the pressure p(P t l) at the point P in the direction I. Thus 



Pressure is hence a force per unit area, as in 1-03(13). 

Now the important points follow: First, experiment and theory show 
that p(P,l) has the same value in each direction /. Thus p depends on 
P alone. Second, since our systems are infinitesimal in size, and since 
they are in equilibrium, the variation of p with P is negligible through- 
out one system. 

We are thus able to define the pressure pofa system as the common value 
of p(Pf) for all points P and directions I in the system. The pressure (in 
mts units) gives rise to a net force of p mts force units normal to the 
boundary surface, per square meter of boundary surface. 

Although the infinitesimal variation of p throughout a system is negli- 
gible in so far as the value of p is concerned, the pressure gradient or rate 
of change of p with respect to distance across the system is of vital importance 
in dynamics. The pressure gradient may assume a large value, being the 
quotient of two infinitesimals. This will be discussed in chapter 4. 

2*05. Temperature. The rigorous definition of temperature is one 
of the major results of thermodynamics, rather than being a simple 
presupposition of this branch of physics. Thus a logical treatise on 
thermodynamics will omit temperature until well into the book, and 
then introduce it by a theorem. (See 2-28.) Such a treatment is 
bewildering to the ordinary student, and we prefer to follow a less rigor- 
ous but more convincing procedure. 

We start from the evidence of our senses that we may distinguish 
between warm and cold bodies. Experiments reveal that when a warm 
and a cold body are put in contact, the former gets colder and the latter 
gets warmer. This continues until a state of thermal equilibrium is 



11 Section 2-05 

reached, by which we mean that there is no further flow of heat. In 
thermodynamics we shall discuss only systems which are in thermal 
equilibrium. 

Any substance which may be brought into thermal equilibrium with a 
mixture of ice and pure water at a pressure of 1 atm (see 107) is said to 
have the temperature centigrade (0C). Any substance which may 
be[brought into thermal equilibrium with steam immediately over water 
boiling at a pressure of 1 atm is said to have the temperature 100C. 
No other temperatures are yet defined. 

Now consider any gas which will not liquefy in the following experi- 
ment. Let its pressure be kept constant at any fixed value p Q . Let its 
specific volume at 0C be a - Let aioo be its specific volume at 100C. 
It will be found that ioo > o- 

When the specific volume has any other value a t , we will define the 
centigrade temperature t (0C) by the linear interpolation formula 

(1) /- 100 -- 1:: -^-- 

<*100 ~ a O 

Of course, this makes temperature dependent on the gas used. All 
that can be said in this treatment is that, for the " permanent gases " 
like helium and hydrogen, the temperatures so defined are consistent to 
within a very small error. This gives us a very reliable " gas thermome- 
ter " from which a mercury or spirit thermometer may be calibrated. 

It is desirable to introduce an absolute scale of temperature, whereby 
the temperature of a gas is proportional to its specific volume. To do 
this we let 



We find empirically that TQ = 273.18 for all the permanent gases, mean- 
ing that they all have about the same coefficient of expansion I/TV 
We then define the absolute temperature T (K) of a system in terms of / 
by the relation 

(3) r-r + /. 

(The K stands for Kelvin.) Let OLT be the specific volume at the abso- 
lute temperature T of the gas in the experiment above (i.e., aj a t ). 
Then from (3), (2), and (1) we get 

OLT 

T=T Q t or 

OQ 

(4) 



Section 2-05 12 

Equation (4), representing the proportionality of a. and T at the fixed 
pressure po is called Charles's law. 

2-06. Meteorological temperature scales. The exact value of the 
TQ of 2-05 requires careful experiment, and as a result physicists have 
changed the accepted value from time to time. To standardize the 
usage in meteorology, which does not require greater accuracy, it is 
customary to use the relation 

(1) 0C=273K. 

This practice will be followed throughout this book. The symbol T 
will often be used for 273 (K). 

It is presumed that the reader is familiar with the Fahrenheit scale 
and knows how to convert it to centigrade by the relation 

/, (F) = I/ (C) + 32. 

Every meteorologist using Fahrenheit on synoptic maps should know 
the following corresponding values, or a similar table: 

/ (C) -40 -10 10 20 30 37 100 

// (F) -40 14 32 50 68 86 98.6 212 

In dimensional formulas we shall use 

(2) [temperature] = [0]. 

2-07. Equation of state. In 2-05 we observed that absolute tempera- 
ture T is measured by the volume change of a suitable substance. The 
reason why this is possible lies in a fundamental property of any of the 
thermodynamical systems of 2-01. 

This property is that between the physical variables p, a, and T 
defining the state of a system there exists a functional relation. The 
relation may be written 

(i) /(p,,r) - o, 

and is called the equation of state of the system. It may be determined 
empirically for real systems to any obtainable degree of accuracy, or it 
may be prescribed for an idealized system like a " perfect gas." 

Except at certain transition states the equation of state can be used to 
determine the value of any one of the physical variables from the values 
of the other two physical variables. 

2 -08. The perfect gas. The so-called permanent gases follow to a 
close approximation a number of well-known laws. One of these is 
Charles's law, given by 2*05(4). Another is Boyle's law, which says that 
at a constant temperature the pressure p and specific volume a are 



13 Section 2-09 

related by the formula 

(1) pa = const, 

where the value of the constant depends on the temperature. Two 
other laws, known as Avogadro's law and Dalloris law, will be stated in 
2-10 and 2-12. 

We define a (thermally) perfect gas as a gas which obeys Charles's and 
Boyle's laws exactly. No such gas exists, but the purely gaseous sys- 
tems considered in meteorology (see 2-01) are so nearly perfect that it is 
most convenient to treat them as perfect gases. 

2 -09. Equation of state of a perfect gas. We shall show that a per- 
fect gas has an equation of state of form 

(1) pa=RT, 

where R is the specific gas constant, which depends on the particular 
perfect gas considered. To balance dimensions we must have 

[R] = [L 2 T- 2 0~ 1 ]. 

Thus from table 1-05 it is seen that R is measured in kj t" 1 deg"" 1 in the 
mts system. 

To prove (1), let a = a(T,p) be the specific volume at temperature T 
and pressure p. Let T be a fixed temperature, and let p Q be the fixed 
pressure of the gas as in 2-05. By Charles's law, 2-05(4), 



By Boyle's law, 2-08(1), 

(3) P*(T,p) - 

Eliminating <*(r,/? ) between (2) and (3), we get 



fA\ ^ /T^ . 

(4) P<*(T,p) ----- - - T. 

^o 

Writing a(T,p) as simply a, and letting R stand for poa(TQ t p Q )/To t we 
get (1), as desired. 

From (4) we also see that, to determine the numerical value of R, we 
need only measure a at say TQ = 273K and po = 101.33 cb. Then 



For example, dry air is considered a perfect gas in meteorology. At 



Section 2-09 14 

/>o - 101.33 cb and TQ = 273K, measurements on dry air give for its 
specific volume a - 

(5) - 773 m 3 t" 1 (1 atm, 273K). 

Then the specific gas constant Rd for dry air is: 



2-10. Molecular weights. The numerical value of the specific gas 
constant for each perfect gas can be obtained directly from considera- 
tions of molecular weight, without actually measuring a(jTo,po) f r 
the individual gas. 

Each pure gas has assigned to it in chemistry a pure number called the 
molecular weight, denoted by m. For our purposes the molecular weight 
may be thought of as simply a relative density at uniform pressure and 
temperature, based on 32.000 for oxygen. The molecular weights used 
in meteorology are given in tables 240 and 2-13. 

TABLE 2-10 
MOLECULAR WEIGHTS 

Helium 4.003 

Hydrogen 2.016 

Water vapor 18.016 

If m denotes the molecular weight of a given gas, then m tons of the gas 
constitute a ton mole, with dimension [M]. The volume occupied by a 
ton mole is called the molar volume, and is denoted by v, with dimen- 
sions [M^L 8 ]. 

An empirical law called Avogadro's law states that at a fixed tempera- 
ture and pressure the molar volume, within a close approximation, is 
the same for all permanent gases. At PQ = 1 atm and TQ = 0C, the 
molar volume is denoted by VQ : 

(1) VQ = 22,414 cubic meters (ton mole)"" 1 . 

The value (1) is assumed exact for perfect gases. 

211. Universal gas constant. Consider a mass of M tons of any 
perfect gas, at any pressure and temperature, occupying the volume V. 
Its specific volume a is V/M. From the equation of state 2-09(1), we 
get 

(1) pV=MRT. 



15 Section 2- 12 

In particular, if the mass is one ton mole, i.e., m tons, then the volume 
is v and (1) takes the form 

(2) pv - mRT. 

Letting p Q - 1 atm and T Q = 0C, we have 

T>T- n P<flQ 

PQVQ mRiQ or mR = ~~ * 



where VQ is given by 2' 10(1). Hence mR = po^o/^o has the same value 
for all gases. It is called the universal gas constant and is denoted by R*. 
R* has the dimensions [L^T^G"" 1 ], and in mts units has the value 



(101.33) (22,414) 

(3) R = ~ 273.18 

Since R* = mR, we get the formula for the specific gas constant for any 
perfect gas in terms of the molecular weight m of the gas: 

1 

(4) R = R* kj t deg . 

m 

This is the formula used to get the numerical value of jR for the equation 
of state 2-09(1) of any pure gas. 

2-12. Mixtures of perfect gases. Dry and moist air are both mix- 
tures of several gases, each of which is treated as perfect. Therefore 
we must learn how to get the specific gas constant for mixtures. 

Let a mixture of volume V cubic meters contain MI tons of gas 1, 
M 2 tons of gas 2, , M 8 tons of gas 5. Let the total mass be M M k - 
Let the respective molecular weights be m\ t m%, , m s . Let the respec- 
tive specific gas constants be RI, R 2 , , R 8 , where each R k = R*/m k , 
according to 2' 1 1 (4) . Assume each constituent is perfect. 

According to a fourth empirical law, Dalton's law, each individual 
constituent gas will obey its equation of state as though the other con- 
stituents were not present. Let p\, p^ ' p 8 be the partial pressures 
of the constituents. Then by Dalton's law and 24 1 (1 ), 

(1) p k V=M k R k T, (t-1, 2, ... f s). 

The total pressure p of the mixture is given by p = Y,Pk- Summing 
equation (1) from k = 1 to 5, we get 

(2) 



Section 2- 12 16 

Now, if we pick R such that 

(3) MR - M k R k , 



then by (2), 
(4) 



MRT. 



But (4) is simply 2-11(1) over again. Thus we have the rule that if we 
define R according to (3) above, then a mixture of perfect gases will also 
have the equation of state of a perfect gas. The formula (3} says in words 
that R is simply a weighted average of the Rk$, each R k being weighted 
according to the mass of gas k present in the mixture. Thus we say 
that the specific gas constant is mass-additive in mixtures. 

2-13. Molecular weight of dry air. The composition of dry air 
varies only slightly. Table 2-13 presents the computation of the 
specific gas constant R = Rd of dry air by the method of 2-12. In the 
first column are the principal constituents of dry air. In the second 
column are their molecular weights (mk). In the third column are their 
individual specific gas constants (R k ) as computed from 2-11(4). In 
the fourth column are the masses (Mk) of the constituents in one ton of 
dry air (so that M =* 1). In the fifth column are the values of M k R k , 
the total of which is equal to Ra, according to 2-12(3). The computa- 
tion shows that, to four figures, Rd = 287.0 kj t""* 1 deg" 1 , in accordance 
with 2-09(6). 

TABLE 2-13 



GAS 


MOL. WT. 
(*) 


Nitrogen 
Oxygen 
Argon 
Carbon dioxide 


28.016 
32.000 
39.944 
44^010 



GAS CONST. PART BY MASS 



296.74 
259.80 
208.13 
188.90 



0.7552 
0.2315 
0.0128 
0.0005 



M k R k 

224.10 

60.14 

2.66 

0.09 



Dry air 



1.0000 = M 286.99 = R d 



It was shown in 242 that a mixture of perfect gases is a perfect gas. 
The'Vnolecular weights of the constituents have been used here only as 
relative densities, and not as relating to the structure of the gas mole- 
cules. It is thus permissible to define a " molecular weight " for a 
mixture, if we choose. The defining relation is 24 1 (4), in order that the 
molecular weight may still be a relative density. 



17 Section 2- 14 

Thus meteorologists define the molecular weight ma of dry air by 

R* 8313.6 ^ ^ , 

(1) ma = = ~^r7T = 28.97 (pure number), 

/v^ 287.0 

With this definition, we have from 2-09(1) for dry air that 

(2) /*--J-**r, 

just as for any other gas of molecular weight m, 

(3) pa - - #*7\ 

w 

With this definition we may treat dry air as though it were one " sub- 
stance " with the molecular weight m&. In particular, we may treat 
dry air as a " pure gas " by the method of 2-12, whenever it is in turn 
mixed with water vapor or other pure substances. 

The reader may prove for himself that in the notation of 242 the 
molecular weight m of a mixture is given by the formula: 

(4) M-= LA* 

m * =1 \m k ) 

Thus the reciprocals of molecular weights are mass-additive in mixtures. 

2-14. Work in thermodynamics. The definition of work in 1-03(14) 
is more precisely formulated in mechanics as follows. When a material 
particle under the action of a force F moves through the distance ds in 
the direction of the force, the work dW 
done by the force is Fds. When the P 

direction of movement makes an angle 
with the force, only the component 
F cos of F in the direction of the 
motion contributes to the work, and 
the work is given by the expression 

(1) dW^ Fds cose. FIG. 2-14. 

When the system considered is an infinitesimal element of fluid, the 
only force with which this element can do work upon its environment is 
that arising from the pressure p on its surface. On each area element dA 
of the surface there is a force pdA pushing normal to the surface, by 2-04. 
Suppose that under the action of the force, the surface element dA 
moves a distance ds in a direction making the angle with the normal 
direction, thus arriving at a new position bA' (see fig. 2-14). Then by 
(1) the work done by the force acting on dA is equal to pdAds cos 0. 




Section 2- 14 18 

It is seen that &Ads cos is equal to the volume swept out by the 
motion of the area element 5A to its new position 5A'. Consider now 
the work d W done by the pressure force in the expansion of a system of 
volume F, surface area A, and mass M. Let dV be the total change in 
volume of the system, being the sum over the complete area A of the 
cylinders swept out by the area elements bA mentioned above. We 
see that 

(2) dW=pdV. 

We shall usually use capital letters to denote quantities referring to 
the total mass of a system, and the corresponding small letters to denote 
the value of the same quantities referred to unit mass (specific quantities). 
Dividing (2) by the mass M, we obtain the expression for the specific 
work dw done by the system: 

(3) dw = pda. 

Our sign convention is sucjhi that if an element expands under its pres- 
sure forces, i.e., does work on the environment, then dw is positive. If 
the element is compressed by the external pressure force, i.e., has work 
done on it by the environment, dw is negative. A system unchanged in 
volume can do no work of the type considered here. 

2 IS. (a,-/>)-diagram. A convenient diagram for many purposes in 
thermodynamics is the (a,-)-diagram, which is a graph of pressure 
against specific volume, both variables having linear scales. On account 
of its frequent application to atmospheric problems, where pressure 
variations are mainly due to vertical displacements of an air element, the 
diagram is drawn with pressure increasing downward. Consider any 
one perfect gas. Its state (see 2-02) is defined by any two of the vari- 
ables />, a, T the third being obtained from the equation of state 
2-09(1). Each point on the (a, )-diagram represents by its coordi- 
nates a unique pair of values of a and p. It consequently represents a 
unique state; conversely, each state is uniquely represented by a point 
on the diagram. 

Any change of state from (a,p) to (a + da,p + dp) is called an ele- 
mentary physical process and is, of course, represented by an infinitesimal 
line on the (a,-)-diagram. A finite process is composed of a succession 
of elementary ones, and is thus represented by a continuous curve on the 
diagram the path of the process. 

Let a gas perform an arbitrary process represented in the (a,-p)- 
diagram by the curve DD'E (fig. 2-lSa). The specific work dw per- 
formed during the elementary process of expansion DD f is pda, which is 



19 



Section 2-16 



measured by the area of the dotted strip. For the finite expansion 
process D to , the total specific work w is equal to the integral of 
2-14(3): 



(D 



w > 






pda. 



This is measured by the whole shaded area under the curve DE. 

Of particular importance is the cyclic process where the system returns 
to its initial state (fig. 2-156). While expanding (DUE) the element 
does positive work. During the subsequent compression (EGD) work 





a -* 
FIG. 2-15a. 



(X * 

FIG. 2-156. 



is done on the element and is negative. It must hence be subtracted. 
The net work done by the element in the complete cycle is therefore 
equal to the area A enclosed by the path representing the cycle in the 
(a, -/>)-diagram. Denoting the cyclic path by c we have 



(2) 



pda=* A. 



The unit of area on the diagram must of course be that of a rectangle 
whose base is the length of a unit (m 3 t"" 1 ) of a and whose height is the 
length of a unit (cb) of p. Multiplying the dimensions of pressure by 
those of specific volume, we see that area on the (e*,-/>)-diagram has the 
dimensions [L 2 T~ 2 ] of specific work. From (2) we have the rule: 

The work performed by unit mass in a cyclic process equals the area 
enclosed by the path in the (a.,-p)-diagram; the work is positive when 
the cycle is taken counterclockwise, and negative when the sense is clockwise. 

246. Isotherms of a perfect gas. As an example of a process repre- 
sented on the (<*,-)-diagram by a curve, we consider the isothermal 
process of a perfect gas. For a particular gas at any constant value T\ 
of r, the product RTi is a definite, known constant. 



Section 2-16 



20 



Then the equation of state 2-09(1) becomes 



(1) 



pa = 



const, 




<x - 
FIG. 2-16. 



whose graph is a rectangular hyperbola in the (a,-p) -diagram (fig. 
2-16). The hyperbola is called the isotherm T = TI. It passes through 
just those points (a,p) which represent states with temperature TI. 

In fig. 2-16 there is also drawn the 
isotherm T = 7\ + dT. 

Now suppose a system is in the 
state represented by the point A in 
the figure. Suppose it is desired 
to heat the system from temper- 
ature TI to TI -{- dT. This may be 
done by any of an infinite number 
of processes, each of which can be 
represented by a line segment start- 
ing from A and ending on the iso- 
therm TI 4- dT. 

We have here pictured just two of these processes. AB is a process 
taking place at constant pressure. This is called an isobaric process. 
AC is a process taking place at constant volume. This is called an 
isosteric process. Note that in the process A C no work is done. In the 
process AB, work is done by the system. This will be of significance in 
the evaluation of the specific heats in 2-21. 

2*17. Heat. It was mentioned in 205 that when two systems at 
different temperatures are brought in contact, the warmer gets colder, 
and the colder gets warmer. A calorimeter is a standard system with 
which different bodies are brought in contact in order to compare their 
various temperature changes with those of the calorimeter. Let the 
calorimeter have the initial temperature 7\, and let a body to be tested 
have the initial temperature T 2 > TI. Let the final temperature of the 
combined systems be. T'. Then TI < T 1 < T 2 . Experiments with the 
same body under varying temperatures show that the final temperature 
T' is determined invariably by the same equation : 

where C and C w are constants. 

Comparing different masses of the same substance in the same calorim- 
eter, it is found that C w remains unchanged, and that the constant C 
is proportional to the mass M of the body : C = cM, where c is a constant 
for the substance. The equation (1) has the form of an equation of 
conservation, in that the term relating to the body is equal but opposite 



21 Section 2- 18 

in sign to the term relating to the calorimeter. The equation suggests 
that there is something which does not change in the process, and which 
flows from the hotter to the colder body. This something is called heat 
and will be denoted by A//. Both terms of (1) have the form 



(2) 

where T 1 is the final temperature. The constant C is called the heat 
capacity of the system. 

Modern measurements have verified the fact that there is conserva- 
tion of heat in all processes of thermal conduction. However, it turns 
out that the heat capacity is not strictly constant, but rather depends 
upon the temperature interval. It is therefore defined by an infinitesi- 
mal process. When dH is the amount of heat required to raise the 
temperature of a substance from T to T + dT, we define the heat capacity 
C at the temperature T as the ratio 

It was mentioned that C - cM. The quantity c is the heat capacity per 
unit mass, or specific heat of the substance. Let dh = dll/M stand for 
the heat imparted per unit mass. Then from (3) : 

dh 



. The accepted unit of heat in physics is the 75 gram calorie 1 cal, 
defined as the heat required to raise the temperature of one gram of pure 
water from 14.5C to 15.5C. We shall, however, always express heat 
in mts mechanical units of energy (see 2*18). 

218. The first law of thermodynamics. The concept of mechanical 
energy and its conservation was established by Leibnitz (1693). He 
showed that in an isolated system the sum of the potential and kinetic 
energies is constant. If a system is not isolated, any loss (or gain) of 
energy is compensated for by the accomplishment of an exactly equiva- 
lent amount of work by (or on) the system. The conservation^of heat in 
all processes of thermal conduction, as formulated in 2-17, was estab- 
lished about seventy years later. 

The role of the first law of thermodynamics is to bring these two sepa- 
rate kinds of conservation into one statement by asserting that mechani- 
cal energy and heat are equivalent to each other and interconvertible. This 
law was first suggested by Count Rumford (1798). However, the credit 
for having set the principle of the conservation of energy upon a firm 



Section 2- 18 22 

experimental foundation is due to Joule. In a classical experiment in 
1849, Joule produced heat by churning water and other liquids with 
paddle wheels and thus determined directly the mechanical equivalent of 
heat. He found that 

(1) lcal = 4.185 x l(T 3 kj, 

and the joule was named in his honor. S6guin (1839), Mayer (1842), 
and particularly Helmholtz (1848) are regarded with Joule as also being 
founders of the first law, because of their important contributions to the 
understanding of its fundamental physical significance. 

Since, according to the first law, heat is equivalent to mechanical 
energy, we shall always express heat in kilojoules, with dimensions 
[ML 2 T~ 2 ]. The conversion to calorie units can always be accomplished 
with (1), if desired. 

We shall apply the first law to the thermodynamical systems of 2-01. 
We shall determine the fate of an infinitesimal amount of heat dll intro- 
duced into the system from its environment. Since we are dealing with 
systems which are in equilibrium, there is no conversion of dll into ki- 
netic or potential energy. The heat dH will in part cause the system 
to expand, and thereby do the work dW against external pressure forces. 
In part the heat will be used to raise the temperature of the system, and 
perhaps also to overcome the resistance of inner forces of attraction 
between the molecules. This second portion will be denoted by dU 
and is called the change of internal energy of the system. The first law 
says in symbols that 

(2) dII=dU+dW. 

Equation (2) is called the energy equation and is the complete mathe- 
matical description of an elementary process performed by a system in 
equilibrium. 

The internal energy U is a measure of the random molecular excita- 
tion of the system. Its value is found to depend only on the state of 
the system ; that is, U is a function only of the mass and of the physical 
variables p, a, and Tt 

If we divide each term of (2) by the mass of the system, we obtain the 
energy equation in the form 

(3) dh = du + dw = du + pda. 

Here u is the internal energy per unit mass; dw is replaced by pda from 
2-14(3). Each term in (3) has the dimensions [L 2 T~~ 2 ] of specific energy. 

2*19. Specific heats of gases. The specific heat of a substance was 
defined in 2*17 as the ratio dh/dT. This definition is adequate for a solid 
or liquid. For a gas, however, we must proceed carefully. Let a gase- 



23 Section 2-20 

ous system at temperature T\ be in the state represented by the point A 
in fig. 216. As remarked in section 2-16, there are an infinite number of 
processes whereby the system can be warmed to the temperature 
TI + dT. Each one requires the absorption of a different amount of 
heat dh. Thus each process defines a different specific heat dh/dT. 

From this multitude we select two specific heats of particular practical 
interest: (i) the specific heat at constant volume (c v ), defined by the iso- 
steric process (da = 0) AC of fig. 2-16; (ii) the specific heat at constant 
pressure (c p ), defined by the isobaric process (dp = 0) AB of fig. 2-16. 
Thus we have 

dh\ dh\ 

1 



(1) c v = 

The dimensions of specific heat are [L 2 T~ 2 9~ 1 ]. 

The two processes selected here must each satisfy the energy equation 
2'18(3). This will lead in the next sections to several relations among 
the specific heats and the internal energy. ' 

2*20. Internal energy of a perfect gas. The (specific) internal 
energy u is a function only of p, a, and T. By means of the equation of 
state 2-09(1), we may eliminate any one of these three variables, for 
example p. Then u becomes a function of two independent variables T 
and a, and may be treated by the calculus as such. The change of in- 
ternal energy du from any given state may always be expressed in terms 
of the changes da. and dT from that state. In the notation of the calcu- 
lus: 



The symbol du/dT is here interpreted to be the rate of change of u with 
respect to jf, in a process for which a is constant. Introducing the 
expression (1) into the energy equation 2-18(3), we get 

(2) dh = ^ dT+ (^ + ^ da . 

This equation is valid for any process. Specializing to the case of the 
isosteric process (da = 0), we obtain immediately from (2) and 2-19(1): 

> * 



Comparing (1) and (3), we see that 
(4) du 



Oa 



Section 2-20 24 

To evaluate du/da, i.e., to see how internal energy varies with a change 
in volume, we must have recourse to experiments. A suitable experi- 
ment is the so-called " expansion into the void." Two vessels, one of 
which contains the gas under high pressure, the other evacuated, are 
placed in communication by means of a pipe uith a stopcock. The gas 
will then rush into the empty vessel without doing work dw, since it is 
pushing against no external forces. If the whole system is insulated, no 
heat dh is imparted to the system. Applied to this process, the energy 
equation dh = d u 4- dw gives du = for each step of the process. Denot- 
ing the finite changes during the complete expansion with a A, we can 
write by means of (4) : 

, du du AT 

(5) Aw =* c v Ai 4- Aa = or = -c v 

Oa Oa Aa 

Joule performed this experiment and found A7"= for all gases used. 
That is, there was no temperature increase during the expansion Aa. 
He thus concluded from (5) that du/da = for all gases. 

Later experiments permitting more accurate measurements showed an 
observable temperature change (Joule-Thomson effect). From such 
experiments, du/da is found to be small. The more nearly a given gas is 
" perfect " in the sense of 2-08, the nearer du/da is to 0. // is therefore 
logical to include in the definition of a perfect gas the stipulation that its 
internal energy be entirely independent of volume, or 

du 

(6) ^ - 0. 

Combining (6) with (4), we obtain 

(7) du - cJT. 
Now by (3) and (6) 

which shows that c v is a function of temperature alone. Experiments 
show that the variation of c v with temperature is the smaller, the nearer 
the gas approaches a perfect gas. A third and last requirement of a per- 
fect gas, therefore, is that c v be constant. 

The integral, u = c v T+ const, of (7) exhibits explicitly the functional 
dependence of u on the state (p, T, a), but (7) itself is sufficient for our 
purposes. 

2*21. Specific heats of a perfect gas. From 2-18(3) and 2-20(7), we 
may write the energy equation of a perfect gas in the form 

a) a- 



25 Section 2-22 

a very useful equation expressing the heat added in terms of the varia- 
tion of the independent variables T and a. In order to compare the 
specific heats c p and c vt we need an expression for dh in terms of the 
variation of the independent variables T and p. To get this, we differ- 
entiate the equation of state pa = RT, whence 

(2) pda - RdT - adp. 
Substituting for pda in (1) we obtain 

(3) dh - (c v + R)dT - adp. 
But then we have for isobaric processes (dp = 0) : 

J = c v + R. 



Comparing this with the definition 2-19(1) of c p , we see that 
(4) c p = c v -f R or c p c v = R. 

These equations (4) say with 2-20 that for a perfect gas, c p and c v are both 
constants j whose difference is equal to the specific gas constant R. It 
will be noted from 2-09 and 2-19 that equations (4) balance dimension- 
ally. 

That c p is larger than c v was already clear in 2-19, because in heating 
at constant pressure the gas must expand, thereby doing external work. 
The heat then is only partly used to raise the temperature. Thus more 
heat is required to raise it one degree isobarically than isosterically 
(where no work is done). See fig. 2-16. 

For dry air, which is regarded as a perfect gas, we shall denote the 
specific heats by c v d and c pd . From experiments their values are deter- 
mined to be as follows: 



(5) c vd - 71 7 kj t- 1 deg- 1 ; c pd - 1004 kj t" 1 deg" 1 . 

Then by (5), c pd - c vd = 287 kj t" 1 deg" 1 = R d , in accord with 2-09(6). 
These values should be remembered. 

Actually c vd and c pd are found experimentally to vary slightly with 
temperature, but the variation may be disregarded in the atmospheric 
range of temperature. 

2*22. Energy equations in logarithmic form. The energy equation 
is the starting point for most of the meteorological applications of 
thermodynamics. It is therefore desirable to express it in various use- 
ful ways. The basic expression is 

248(3) dh = du + dw. 



Section 2-22 26 

To express dh in terms of the independent variables T and a, we have 
2-21(1) dh=cJT+pda. 

To express dh in terms of the independent variables T and p, we can 
combine 2-21(3) and 2-21(4): 

(1) dh = c p d T adp. 

The last form will be used most often, since the variables T and p are 
those directly observed in the atmosphere. 

The energy equation assumes a convenient form when both sides are 
divided by the temperature T, and the equation of state pa = RT is used. 
Equation 2-21(1) becomes 

(2) ^ = C f + *^' 

1 1 a 

Similarly, (1) becomes 

dh dT dp 
fi\ __ - ~p ___ 

TT~~to 

Now by differentiating the logarithm of the equation of state, called 
logarithmic differentiation, we get 

dp da dT 

W ~r H ~TP 

p a 1 

By means of (4) we can eliminate d T from either (2) or (3). Then since 
by 2-21 (4) c p = c v + R, we get 

f*\ *_ ^ *t 

wJ ^ c p T~ c v 

Tap 

The three forms (2), (3), and (5) of the energy equation may be written 
together in an order which makes them all easy to remember: 

!** _ p d T _ ^L i? ^ ^ *? 
(6) _ == x^ + c v -~- = Cp - R = c v -\- c p 

T a T T p pa 

Note that the variables in (6) are written in a certain cyclic order, start- 
ing with a, r. The constants are written twice in order of magnitude, 
separated by a minus sign : 



The symmetry of (6) is a superficial aspect of a far-reaching physical 
simplification which results from dividing the energy equation by T. 
This will be discussed in 2-28. For the present it suffices to note that 



27 Section 2-24 

the expressions in (6) are differentials of functions of state. Thus we 
have: 



(7) j - d[ln a R T Cv ] - d[\n T c *p~ R ] - d[\n p c a. c }. 

By In x we shall always mean the natural logarithm of x, i.e., the loga- 
rithm with base e = 2.71828 . 

2 '23. Atmospheric processes. The most fundamental process which 
adds or subtracts heat energy to or from a parcel of air in the atmosphere 
is radiation. Other important agencies for the exchange of heat are 
conduction, turbulent mixing, and internal friction. All these proc- 
esses are continually influencing every element of the atmosphere in a 
complicated fashion, usually inaccessible to a detailed thermodynamic 
analysis. However, they all proceed quite slowly, compared with 
another important class of processes, which it will be our primary 
objective to analyze. 

These other processes are those caused by the motion of the air and 
primarily by the vertical motion. They proceed with relative rapidity, 
so that they can profitably be investigated by neglecting the influences 
of the slow processes involving heat exchange between the system and its 
environment. We shall therefore concentrate our attention on proc- 
esses for which there is assumed to be no heat exchange between the 
system and its environment. Such a process is called adiabatic. 

Later, in chapter 3, we shall discuss processes which are not strictly 
adiabatic, but there the heat exchange will be small. 

2-24. Adiabatic processes of a perfect gas. The condition which 
must be satisfied for an adiabatic process is by definition dh = 0. In 
2 22 there are many expressions involving dh and other variables. 
Each of them becomes a differential equation of the adiabatic process of 
a perfect gas, when dh is replaced by 0. For reference, they are repeated 
here: 

(1) du + dw=Q; 

(2) 

(3) 

(4) ^^.f^f-A^- 

a T 1 p p a 

(5) d[ln a R r*} - d(ln T e "p- R ] = d[ln p e >a e >>} - 0. 



Section 2-24 28 

From the integration of (5), we get three equivalent relations between 
the variables of state in an adiabatic process: 

(6) a R T Cv - const; T Cp p~ R = const; p Cv a c *> = const. 
Defining two pure numbers K and rj by the relations 

/*\ -^ j C P 

(7) K - and 77 = > 

p C v 

we can rewrite the more important two of the equations (6) in the form : 

(8) > T = const -p"; 

(9) ^a 17 - const. 

These are the equations derived by Poisson (1823), and generally bear 
his name. For this book, (8) is the more important, and K will be used 
often. Formula (9) will be discussed presently for illustration; the use 
of 17 is not standard in meteorology. Both K and ry have dimension [1]. 

It must be understood that the constants of integration in (8) and (9) 
are fixed for one adiabatic process, but they will have different values for 
other adiabatic processes. They are each determined separately from 
the values of the physical variables at a given initial state, exactly as the 
constant in the equation of a straight line of slope 6, 

y = 6x -f const, 

is determined by knowing one point on the line. 

We may also rewrite (8) and (9) in terms of the values pi, i, 7*1 of 
the variables at an initial state: 

$ 

(11) 

These equations are analogous to the point-slope equation of the straight 
line referred to above : 

y-yi- 6(x-xi). 
For dry air, the Poisson constants K and i\ have the following values: 

(12) K d = - 0.286, ri d - ^ - 1.400. 

Cpd C v d 

These values should be remembered. 

2*25. Adiabats on diagrams. Each adiabatic process, like any other 
process, is represented by a definite curve on the (a,-)-diagram. The 
curve is called an adiabat, and its equation is 2-24(11). The whole 



29 



Section 2-25 



family of adiabats is given by 2*24(9) 

pot 1 = const, 

as the constant varies. The family of isotherms was found in 2-16 to be 
given by the equation 216(1) 

pa = const, 

as the constant varies. The adiabats and the isotherms are distinct 
curves, since r; > 1. The actual shape of the adiabats depends on the 
value of rj. For dry air, when t\ = 17^, they are called dry adiabats. 



100 cb 




700 



2500 



3000 



FIG. 2-25a. (a, p) -diagram. 



In fig. 225a we have an (,-/>) -diagram for the range of values of a 
and p observed in the lower atmosphere. The isotherms T * 200, 300, 
400, and 500 are drawn in solid lines; the dry adiabats 6 - 200, 300, 400, 
and 500 are drawn in dashed lines. 6 is defined in 2*26. The adiabats 
have the same asymptotes as the isotherms, but their slope 
(dp /da = qp/oi) is steeper than the slope (-dp/da = p/a) of the 
isotherms. 

The (a,-p)-diagram is very suitable for a theoretical analysis of 



Section 2-25 



30 



atmospheric processes, since work is measured by an area on the dia- 
gram, as in 2-15(1). However, it is very inconvenient in actual practice 
for several reasons. First, neither the adiabatic nor the isothermal 
processes are represented by straight lines. Second, the variation of the 
variables requires that the adiabats and isotherms meet at too small an 
angle for easy discrimination. See fig. 2-25a. Third, the areas of most 
importance in meteorology are spread out inconveniently on the page, 
making it difficult to design this diagram as a well-shapecj, large-scale 
chart for detailed use. 







I 



10 
20 

40 

60 
80 

100] 







\ 



\ 






0K 100 200 T -^. 300 

FIG. 2-256. Stuve diagram. 



400 



Much better diagrams for practical purposes are those involving p 
and T as coordinate variables. These are more logical anyway, being 
those directly measured. The simplest of these diagrams is the Sttive 
diagram, which is the basic diagram on which the so-called pseudo- 
adiabatic chart is drawn. The Stuve diagram is designed to make the 
adiabats straight lines, while keeping p and T as the coordinate variables. 
This is accomplished by the device of using a linear temperature scale, 
but making the pressure coordinate represent pressures in terms of their 
*dth powers. Thus the isobar p cb will be p Kd units from the axis p - 0. 
The scale is for convenience still labeled in centibars. (See fig. 2-2S&.) 
But now from 2-24(8) it will be seen that each dry adiabat will be a 
separate straight line through the origin (p = 0, T = 0). These are 
drawn in the figure, and finally the portion of the diagram of greatest 



31 Section 2-26 

meteorological interest is drawn in heavy lines and enclosed in a frame. 
The rest is usually omitted from the meteorological charts. 

The fact that in older books the value KA = 0.288 was used explains 
why this value occurs on many pseudo-adiabatic charts. 

2 -26. Potential temperature. The various dry adiabats on the Stiive 
diagram (or any other diagram) need to be labeled. The standard 
method is to label each adiabat by the temperature 6 at which it crosses the 
isobar p = 100 cb. Any parcel of dry air is in some state (p, T, a), which 
is uniquely defined by p and T. If we plot this state on the diagram, it 
will lie on some dry adiabat. The corresponding value of 6 is the tempera- 
ture read off the same dry adiabat at 100 cb. This temperature 6 is 
defined to be the potential temperature of the parcel of air in the state 
(p, !T, a). In physical language the potential temperature is the tem- 
perature which the air parcel assumes when compressed (or expanded) 
adiabatically to a pressure of 100 cb. 6 is always expressed in degrees 
absolute and has the dimension [0] of temperature. 

By 2-24(10) we see that the potential temperature 6 of dry air can be 
computed from the formula 



The student should be able to use formula (1) readily, with the aid of 
logarithm tables, even though in practice 6 is usually estimated from a 
diagram. When is given a constant value, (1) gives the variation of T 
and p in the dry-adiabatic process. It is equivalent in this respect to 
2-24(8). 

The potential temperature is an invaluable aid to the thermodynami- 
cal study of the atmosphere. The main reason for this is that, as 
mentioned in 2-23, short-term atmospheric processes are adiabatic. 
Unless the air is saturated, they are approximately dry adiabatic. But 
in a dry-adiabatic process, 6 remains unchanged, even though T and p 
may change a great deal. Such a quantity, which remains invariant 
under dry-adiabatic processes, is called a conservative property of atmos- 
pheric air. Thus potential temperature is conservative, and as such 
may be used to identify air masses through a short interval of time (say 
24 hours). 

For dry air still another form of the energy equation can be obtained 
by logarithmic differentiation of (1). We get 

dO dT dp dT R d dp 

(2) T == ~T r ~* d T~ =3 ~T r ~~T~T > 

6 T P T c pd p 



Section 2-26 



32 



where 
obtain 

(3) 



has been removed by 224(12). Multiplying (2) by c p d, 



dO dT 

c pd - c pd - 



dp 




Comparing (3) with 2-22(6), we see that 
(4) 



dh dd Jf f _ 

- c pd j = d(c pd lnO). 



By (4) we see that when dh = 0, then dO = (which we already knew 
We can also compute the magnitude of the change dO of potenti; 
temperature, due to the introduction of dh kj t"" 1 of heat. 
// must be emphasized that so far 6 is defined only for dry air. 

2 '27. Differentials and functions of state. A physical variable 
called a function of state if it can be expressed as a single-valued math 
matical function of two of the variables a, p, T defining the state: fc 

example, as a function of a and \ 
In physical language, a variable 
a function of state whenever in tf 
(a,-)-diagram it is possible 1 
draw lines along which the var 
able assumes one constant valui 
Such lines are called isopleths of th 
variable. Some of the functior 
of state so far considered are no 
listed, together with the sped; 
names (if any) of their isoplethi 
P 227 p (isobars), r(isotherms), 0(adu 

bats) , a (isosteres) , p (isopycnics) , i 

The differentials so far considered may be placed in two groups. I 
the first group are dp, dT, dO, da, dp, and du, all of which are differentia! 
of the functions of state just mentioned. The differential of any fun< 
tion of state is called an exact differential. 

In the second group are dw and dh, which are not differentials of fun< 
tions of state, and are therefore called inexact differentials. That is, n 
functions of state w and h exist whose differentials are dw and dh. This wi 
be proved below. 

We first quote a theorem from the calculus: 

(1 ) The integral of an exact differential around a closed path is zero. 

The rigorous proof of (1) may be found in any textbook of advance 
calculus. The general idea may be illustrated by considering f d1 
where c is the closed path 13421 in the (a,-p)-diagram of fig. 2-2' 




33 Section 2-28 

Starting the integration at 1, it turns out that the integral fdT along c 
from 1 to 3 is equal to the net change of T between 1 and 3. This can be 
measured by subtracting the T-value T\ of the isotherm through 1 from 
the jf- value T$ of the isotherm through 3, giving T 3 - 7\. Let the inte- 
gration be continued through points 4 and 2 back to the starting point 1 
again. The value of f c dT will be the net change of T over the complete 
path. This will be obtained by subtracting the T-value T\ from itself, 
giving zero. 

Now let us integrate the energy equation 2-18(3) around a cyclic path c 
enclosing the positive area A. By (1), f c du= 0. Using 2' 15 (2), we 
then have 



(2) 



/dh = / du + / dw - / dw - A > 0. 
Jc Jc Jc 



From (2) , we see that neither f c dh nor f c dw is zero. It then follows from 
(1) that dh and dw cannot be exact differentials, as asserted previously. 

Exact differentials are very handy to deal with mathematically. 
The integral from 1 to 2 of any exact differential d<p may be evaluated as 
the difference <p 2 - <Pi in the values of the ^-isopleths between states 1 
and 2. For an inexact differential like dw, on the other hand, there are 
no isopleths to draw, and J\dw depends entirely on the path of integra- 
tion from 1 to 2. See 2-15. 

2*28. Entropy. Equation 2-26 (4) states that for dry air dh/T is equal 
to the differential of the function of state c pd In 9. From 2-22(7) it is 
seen that for any perfect gas dh/T is the differential of a function of 
state, namely, any of the functions in brackets in 2-22(7), which differ only 
by a constant. According to the definitions of 2-27, in these instances 
dh/T is an exact differential. 

The foregoing are but instances of a very general thermodynamical 
theorem, whose proof is given in any standard textbook on thermo- 
dynamics. This asserts for an arbitrary system, whether or not the 
equation of state and the internal energy function are known, that the 
expression dh/T is always an exact differential. As such it is the differen- 
tial of a function of state which is called the (specific) entropy of the 
system, denoted by s. The dimensions of specific entropy are those of 
specific heat, namely, [L*!*""^"" 1 ]. The total entropy S is equal to s 
times the mass of the system. 

We have always 

, dh [dh 

(1) ds = or 5= / + const. 



Section 2-2S 34 

Being defined differentially, s is known only up to an arbitrary constant. 
For dry air, because of 2-26(4), 

(2) s = Sd = c p d In + const. 
For any perfect gas, because of 2-22(7), 

(3) 5 = In [a R T Cv ] + const = In [T c ^p~ R ] + const - In \p c *cf*] + const. 
In general, for any system 

(4) 5 = some function of a, p, and T. 

Since by (1) an adiabatic process (dh = 0) is a process for which 
entropy is constant (ds = 0), adiabatic processes are often called isen- 
tropic. 

The factor 1/Tis called an integrating factor for the inexact differential 
dhj since dh/T is exact. It is this which yields the logical definition of 
temperature mentioned in 2-05: logically temperature is simply the 
reciprocal of the integrating factor for dh. 

The fact that ds is exact makes entropy very important as a heat 
variable in thermodynamics. Because of its abstract definition, how- 
ever, entropy is rather bewildering to the student, and we shall use it as 
little as possible. For dry air, the concrete meteorological variable 6 
can completely replace entropy. 

229. Thermodynamic diagrams. It has already been shown that 
thermodynamic processes can be represented and studied on a diagram 
whose coordinates are the independent variables of the system. Any 
such diagram is called a thermodynamic diagram, and the first example 
was the (,-/>) -diagram of 2-15 and 2-25. Another was the Stiive 
diagram of 2-25. We shall conclude this chapter with a discussion of the 
emagram and tephigram. 

A great deal of dispute is heard among meteorologists as to which are 
the best diagrams. The majority of problems can be worked theoreti- 
cally with equal facility on all the basic diagrams, since they are all maps 
of each other. What really makes one diagram easy to use is the 
accuracy, clarity, and convenient scale with which its plate has been 
drawn. For certain special problems, special scales are required, and 
these can (usually ) be put on any diagram, but in practice they are found 
only on special prints made for the purpose. 

As for the basic diagrams in frequent use in the United States the 
Stiive diagram, the emagram, and the tephigram it is most important 
that the student first realize their similarities, not their differences. 
The fact is that all are " maps " of each other. Like all maps, all can 
show the same lines, if desired. Finally, like all maps, each bears the 
individuality of the particular projection. 



35 Section 2-32 

230. Important criteria of the diagram. There are three criteria 
which we shall use to examine each diagram after it has been defined. 

(i) How large an angle is there between the isotherms and the adia- 
bats? A large angle is desirable, since soundings drawn on the diagrams 
will be analyzed on the basis of their slopes. The larger the angle, the 
easier it is to distinguish important changes of slope. 

(ii) How many of the important isopleths (isobars, isotherms, adia- 
bats, etc.) are straight lines? The more straight lines and the less 
curved lines, the easier the diagram is to use. 

(iii) Is the work done in a cyclic process proportional to the area 
enclosed by the curve representing the process? This is an essential 
feature in theory, and it is important in practice for certain operations. 
On the whole, however, this feature has probably received too much 
emphasis in meteorology. 

2 -31. Stiive diagram. As it was discussed in 2-25, the Stiive diagram 
need not be defined again. The student should get a pseudo-adiabatic 
chart and make sure he can determine the pressure, temperature, and 
potential temperature of any point on it. The other lines will be intro- 
duced in chapter 3. As for the criteria of 2-30: 

(i) Theoretically, by stretching the T axis sufficiently, the angle 
between adiabats and isotherms in the atmospheric range could be made 
arbitrarily close to 90. In practice, to keep the diagram more or less 
square and legible, the angle is near 45. See fig. 2-256. 

(ii) Isobars, isotherms, and adiabats are all straight lines. 

(iii) The work done is not proportional to the area enclosed but 
depends also on which pressures the area covers. The variation is rather 
gradual. For example, one square centimeter represents about 25% 
more energy at 40 cb than the same area at 100 cb. 

2 32. Emagram. The emagram is a graph of -In p against T. 
It was specifically designed to be a pressure-temperature graph having 
the work-area property (iii) of 2-30. From this property Refsdal gave 
the diagram its name, as an abbreviation for " energy-per-unit-mass 
diagram." It is sometimes also named after Hertz, Neuhoff, and 
Vaisala, who discussed the diagram and added features to it. 

The emagram has a linear temperature scale on the horizontal axis, 
and a logarithmic pressure scale, increasing downward, as a vertical 
coordinate. Since as -0, lim(~ln /?)= oo, the diagram must in 
practice be cut at some low pressure, usually 4 cb. 

After the axis scales have been drawn (see fig. 2-32a) the adiabats are 
drawn in by formula 2-26(1), or else they are plotted from a Stiive dia- 
gram. The equation of any adiabat can be obtained by taking the 



Section 2-32 



36 



logarithm of 2-26(1) for 6 - const: 



(1) 



In p = - In T + const. 
"d 



Formula (1) shows two things. First, each adiabat is a logarithmic 
curve on the emagram, becoming steeper with decreasing T. Second, 




100 200 300 

r 

FIG. 2-32a. Emagram. 



400 



any two adiabats may be brought into coincidence by a displacement 
parallel to the In p axis, so that all adiabats are congruent. As for the 
criteria of 230: 

(i) As in the Stiive diagram, the angle between adiabats and iso- 
therms in the atmospheric range can be adjusted to any value short of 
90. But again the convenience of scale and economy of paper dictate 
that the angle be about 45. 

(ii) The isobars and isotherms are straight lines, but the adiabats are 
logarithmic curves. 

(iii) The work w done in a cyclic process c is proportional to the 
area A 9 enclosed by c on the emagram. To see this, we recall from 



37 



Section 2-32 



2-21 (2) that 

dw pda 
Hence the work w is given by 

(2) ldw = iRddT- \adp= - / adp, 

Jc Jc Jc Jc 

where one integral is by 2-27(1). Now a = RdT/p by the equation of 
state. Hence from (2) 



(3) 



w = - 



P 



[ 

Jc 



The last equality follows for the emagram, whose coordinates are T and 
In />, just as 2'15(2) is true for an (a, p) -diagram. 



20^4) 




200 220 240 260 280 300 320 



FIG. 2-326. Emagram. 

One other feature of the emagram as drawn in practice must be men- 
tioned. Note in fig. 2-32a that the isobars for p = 4, 8, 12, 16, and 20 cb 
appear to have exactly the same relative position and spacing as the 
isobars whose pressures are respectively five times as large, namely, 
p = 20, 40, 60, 80, and 100 cb. Thus the portion of the coordinate grid 
on the emagram between 4 cb and 20 cb appears to be congruent to that 



Section 2-32 38 

portion between 20 cb and 100 cb. If the graph were cut in two at 20 cb 
and the two halves superimposed as in fig. 2-32, then each isobar would 
represent two pressures: (i) a " high " pressure above 20 cb; (ii) a 
11 low " pressure under 20 cb, which is one-fifth of the corresponding high 
pressure. 

The reason for the congruence of the high- and low-pressure scales is 
the logarithmic pressure scale. To draw a logarithmic scale, a base line 
is fixed, corresponding to the isobar p = 1 (In p = 0). A unit of length 
is fixed. The isobar for p cb is drawn In p units away from p = 1 . Now 
In 5p = In 5 -f In p. Hence the isobar for 5p cb is just In 5 -f In p units 
away from p = 1. Hence the isobar for 5p cb is found just In 5 units 
below the isobar for p cb. This results in the congruence mentioned. 

Not only the isobars do double duty in fig. 2*326, but the adiabats also 
can serve with either the high pressures or the low pressures by simply 
changing the value of their label 0. This follows from the fact that all 
adiabats are congruent to each other. 

Fig. 2 -32b is drawn with the low pressures in parentheses. The values 
of to be used with the low-pressure scale are also in parentheses. The 
high pressures and corresponding values of 6 are without parentheses. 
The diagram has also been cut to the atmospheric temperature range. 
The student should get an emagram and become familiar with the scales 
introduced here. 

Exercise. Prove that if an adiabat has the value 8 = 61 on the high-pressure 
scale, then it has the value = 5** 0i on the low-pressure scale. Check fig. 2-32& 
by this formula. 

233. Tephigram. The tephigram is a graph of In against T. 
This diagram has a linear temperature scale and a logarithmic 6 scale 
which is a linear entropy scale, by 2-28(2). The diagram was adopted 
for meteorological use by Shaw (who used the symbol </> for entropy), 
and was called by him a T-<t>-gram or tephigram. The coordinates are 
shown in fig. 2-33a. 

The equation for the isobars in terms of T and as independent vari- 
ables is given by 2-26(1). For each isobar, p is constant, whence 
- const r. Thus, 

(1) In 6 - In T + const (isobar). 

From (1) we see that each isobar is a logarithmic curve on the tephi- 
gram. Furthermore, all isobars are congruent, and any two isobars may 
be brought into coincidence by a displacement parallel to the In axis. 
See fig. 2-33a. Equations 2-32(1) and 2-33(1) may be compared. 

In fig. 2-33a the approximate range of variables commonly used is out- 



39 



Section 2-33 



lined with a rectangle. This rectangle is then rotated, so that the iso- 
bars are roughly horizontal. A sketch of the resulting diagram is shown 
in fig. 2-336. As for the criteria of 2-30: 

(i) The angle between the adiabats and isotherms is exactly 90, 
since these lines are the coordinates of the tephigram. This 90 angle is 
probably the greatest advantage of this diagram. 




i-200 



LlOO 

0K 100 200 300 

FIG. 2-33a. Tephigram. 



FIG. 2-336. Tephigram. 



(ii) The isotherms and adiabats are straight lines, whereas the iso- 
bars have a curvature which is slight in the atmospheric range. 

(iii) The work w done in a cyclic process c is proportional to the area 
A" enclosed by c on the tephigram. To see this, note from 2-26(4) 
that 



(2) 



dh - c pd Td(ln 0). 



Now by (2) and 2-27(2), 

(3) w = / dw B / 

J c J c 



</. 



dh - c pd / Td(ln 0). 



But the last integral is c p ^A h \ whence w is proportional to A ". 

The student should get a tephigram, and become completely familiar 
with the coordinates so far introduced. Above all, he should see that the 
Stiive diagram, emagram, and tephigram are all minor distortions of 
each other. 



CHAPTER THREE 



THERMAL PROPERTIES OF WATER SUBSTANCE AND 

MOIST AIR 

3 -01. Isotherms of water substance. The preceding chapter has 
treated the thermal properties of dry air in equilibrium as a special case 
of a perfect gas. Any real gas actually behaves nearly like a perfect 
gas in a temperature range where it can neither liquefy nor solidify. 
The isotherms of a real gas are therefore nearly rectangular hyperbolas 





or *- 
FIG. 3-Ola. 



FIG. 3-016. 



in the (a ,p) -diagram. (See fig. 3-Ola and compare fig. 2-16. Here we 
follow the practice of physicists in having pressure increase upward. 
This is convenient, since the system considered at present is not the 
atmosphere. Cf. 245.) 

Water substance, however, does liquefy and freeze in the atmosphere. 
Its isotherms are therefore complicated. Consider a sample of pure 
water vapor in a cylinder. We shall denote the vapor pressure by e, 
reserving p for pressure in the atmosphere. Let the vapor be com- 
pressed by a piston while the system remains at a constant temperature 
of say 300K. We shall follow the true isotherm 300K on an (a.e)- 
diagram, as drawn from empirical evidence. See the schematic diagram 
in fig. 3-016. 

The specific volume a will roughly follow the perfect gas behavior from 
A until the vapor becomes " saturated " at a definite pressure, depend- 
ing on the temperature. This pressure is called the saturation vapor 

40 



41 



Section 3-02 



pressure, and in the present instance equals 3.6 cb (see point B). Fur- 
ther compression by the piston does not change the pressure e. Instead, 
the vapor gradually condenses, and the isotherm proceeds from B 
(a 39,000 m 3 t" 1 ) to C (a 1 m 3 t" 1 ), where we have only liquid 
water. After this, further compression can reduce a but very little, 
since water is nearly incompressible. Thus the isotherm continues from 
C nearly parallel to the e axis to D and beyond. 

If we follow the isotherm 250K in the same manner (see dotted line in 
fig. 3-016), we get a similar pattern, except that no liquid stage occurs. 
The vapor starts to solidify directly to ice at e = 0.077 cb, a 1.5 X 10 6 
m 3 IT 1 (point #') All the vapor is solidified into ice at the point C 1 
(a 1.09 m 3 ^ 1 ). Further compression of the ice at ordinary high 
pressures results in no significant volume change. 

3*02. (a,e)-diagram and the triple state. The above considera- 
tions show that we must examine in some detail the behavior of water 
substance under equilibrium conditions. To a certain extent it is possi- 
ble to describe the liquid and vapor phases of all fluids by one equation, 



Vapor 




T >T C 



T<T 



FIG. 3-02. 

the van der Waals equation of state. Since the van der Waals formula is 
not very successful with water and water vapor, and since it does not 
even pretend to describe the solid phase, we will omit any such discus- 
sion. We prefer to give the (a,e)-diagram obtained from experiments on 
water substance. 

Fig. 3-02 shows a number of isotherms in the (ct,e) -diagram. For 
temperatures above the critical temperature T c (647K for water sub- 
stance), the vapor never condenses, and the isotherms are roughly like 



Section 3-02 42 

those for a perfect gas. For temperatures between the triple state 
temperature T t (273K for water) and T c , the isotherms look like the solid 
line in fig. 3-016. The significance of T t and T c is explained in this and 
the following sections. For T < T tl the vapor condenses directly to ice 
on compression, as shown. On further compression the specific volume 
of the ice remains near 1.09 m 3 t"" 1 , until at high enough pressures the 
ice melts to water, or else changes to a second crystalline form of ice. 
Neither of the last processes is represented in fig. 3-02, as it would unduly 
complicate the left side of the diagram. The ice - water transition will 
be discussed in detail in section 3' 13. 

The empirical evidence thus shows that the (a,e)-diagram is divided 
into several regions, each region representing one type of phase equilib- 
rium for water substance. Some of these regions are outlined with 
heavy lines in fig. 3-02 and are labeled vapor, water, water & vapor, ice 
& vapor. The regions ice and ice & water have been omitted. 

The line A3 represents all states where the three phases ice, water, 
and vapor can exist simultaneously in equilibrium in any relative pro- 
portions. This is the triple state. It is found to occur for water sub- 
stance at 

(1) e= 0.611 cb; T t - T Q = 0.007SC. 

The corresponding specific volumes of ice, water, and vapor are respec- 
tively (in m 3 t" 1 ) 

(2) a; = 1.091; a w - 1.000; a v 206,000. 

One phenomenon has been entirely omitted in this discussion, the 
supercooling of water. The discussion of this topic is reserved i i 
section 3-15. 

3-03. The critical state. The point C in fig. 3-02 is the point where 
the critical isotherm touches the water & vapor region. The correspond- 
ing state is called the critical state. For water substance, the critical 
state occurs at approximately* 

(1) e c = 22,100 cb 218 atm; T C -647K; c - 3.1m 3 IT 1 . 

The critical temperature T c is the highest temperature at which water and 
vapor can co-exist in equilibrium. The critical pressure e c is similarly 
the highest pressure at which water and vapor can co-exist in equilib- 
rium. The critical specific volume a c is the value of a observed at the 
critical temperature and pressure. 

* N. Ernest Dorsey (comp.), Properties of Ordinary Water- Substance, Reinhold 
Publishing Corp., New York, 1940; p. 558. 



43 Section 3-04 

It is seen from fig. 3-02 that it is possible to take a sample of vapor into 
the state labeled water without passing through any transition zone, for 
example, by keeping the pressure greater than e c \ i.e., there is no bound- 
ary curve in fig. 3-02 separating water from vapor. This corresponds to 
the experimental fact that vapor goes into water at these pressures with- 
out any abrupt transition. 

It is arbitrary whether we call the fluid a vapor or a liquid in this 
region. For a. < a c , the critical isotherm is customarily taken as the 
boundary between vapor and liquid. We then have the following rule 
pertaining to all fluids : It is impossible to liquefy a substance at tempera- 
tures higher than the critical. This law explains why the permanent 
gases of the air are never liquefied at atmospheric temperatures: their 
critical temperatures are too low. 

Table 3-03 gives the critical data* for some permanent gases and 
carbon dioxide. 

TABLE 3-03 

GAS T c (K) p c (cb) <x c (m 3 1" 1 ) 

He 5.2 230 14.4 

H 2 33.2 1300 32.2 

N 2 126.0 3390 3.2 

O 2 154.3 5040 2.3 

CO 2 304.1 7400 2.2 

3*04. Thermal properties of ice. Beginning with ice, we shall dis- 
cuss separately the thermal properties of the three phases of water sub- 
stance and the various changes of phase. 

Ice, the solid phase of water substance, is known to exist in several 
different crystalline states, each of which is properly a phase itself. 
Since only one of these phases occurs in the atmosphere, we shall ignore 
the others 

At 0C the specific volume oti of ice is given by 

(1) on- 1.091 m s t" 1 . 

On cooling below 0C ice contracts so slowly that we may regard as 
constant for our purposes. 

The specific heat Ci of ice (cf. 2-17) varies with temperature, but the 
0C valuef 

(2) a - 2060 kj IT 1 deg- 1 

is sufficiently accurate for atmospheric problems. 

* Charles D. Hodgman (edit.), Handbook of Chemistry and Physics, 25th edition, 
Chemical Rubber Publishing Co., Cleveland, 1941; pp. 1703-1705. 

t Edward W. Washburn (editor-in-chief), International Critical Tables of Numeri- 
cal Data, McGraw-Hill Book Company, New York, 1926 ff., 7 vol. and index; vol. 5, 
p. 95. 



Section 3-05 44 

3-05. Thermal properties of water. The specific volume a w of 
(liquid) water assumes its least value ct w = 1.00 m 3 t" 1 near 4C. (This 
value will be recalled from section 1-04 as the definition of unit specific 
volume.) a w increases somewhat at higher and lower temperatures, 
getting as high as 1.043 m 3 t" 1 at 100C. In the atmosphere we shall 
use 

(1) a w = 1.00 m 3 ^ 1 

in computations, regarding a w as constant for practical purposes. 

The specific heat of water is denoted by c w , and it varies slightly with 
temperature. The gram calorie was defined in 247 as the heat required 
to heat one gram of water one degree at 1 5C . Referring to the mechani- 
cal equivalent of heat, 2-18(1), we have at 15C 

(2) c w , = 4185kjt- 1 deg- 1 . 

In our subsequent work we shall treat c w as a constant with this value. 

3 '06. Equation of state of water vapor. As in section 3-01, the pres- 
sure exerted by water vapor will be denoted by e. The other physical 
variables relating to water vapor will be denoted by the subscript v, 
for example, a v . With the assumption that water vapor behaves closely 
enough like a perfect gas, its equation of state 2-09(1) has the form 

(1) ea v =R v T. 



The gas constant R v is computed from the relation 2-11(4), J?* 

where m v (= 18.016) is the molecular weight of water vapor. This gives 

> a-S 

For later work it is convenient to express R v in terms of Rd, using the 
relation R* = mJR v = m d R d , where by 2-13(1) m d (= 28.97) is the 
molecular weight of dry air. Thus 



where 

(4) c = - - - 0.622. 

nid 

By introducing in (1) the expression (3) for R v , the equation of state for 
water vapor takes the form in which we will generally write it: 

(5) ea v = 



45 Section 3-07 

It will be useful later to know the magnitude of a v under certain special 
conditions. The following values are computed from (5). At the 
normal boiling point 

(6) (a v )i atm , 373 = 1699 m 3 t" 1 . 

Saturated vapor at the triple state (nearly 0C) has 

(7) MOMI cb, 273 = 206,200 m 3 IT 1 . 

Saturated atmospheric air seldom is warmer than 35C. The corre- 
sponding saturation vapor pressure (5.62 cb) is therefore the highest, 
and the corresponding value of a v is the lowest, value which normally 
occurs in the atmosphere. This value is 

(8) (a,) 5 .62 cb, 308 - 25,290 m 3 t" 1 . 

As a check on the accuracy of (5), we give the values of ot v in the three 
cases above, as taken from experimental data in the Handbook of Chemis- 
try and Physics:* 

Ml atm, 373 = 1671 m 3 t" 1 } 

(t)o.Gii cb, 273 = 206,300 m 3 t" 1 \- (empirical). 
(<Os.e2 cb, 308 = 25,250 m 3 t" 1 J 

Thus it will be seen that at atmospheric temperatures (5) gives all 
accuracy possible from the data. At the boiling point a 2% error is 
found. For temperatures and pressures approaching the critical, it is 
found necessary to use a more refined equation of state. 

3-07. Specific heats of water vapor. For ice and liquid water the 
specific heats Ci and c w are practically independent of the type of process 
used to heat the substance. For water vapor, however, as in section 
2-19, we must distinguish various specific heats. The specific heat of 
water vapor at constant pressure will be denoted by c pv . The specific 
heat of water vapor at constant volume will be denoted by c vv . 

The variation of c pv and c vv with temperature is quite considerable. 
Furthermore, there are few data available as to their values at atmos- 
pheric temperatures. Simply to fix the magnitudes of these quantities 
for our calculations, we shall regard c pv and c vv as constants in the atmos- 
phere, with the following values:! 

(1) c pv = 1911 kj r 1 deg- 1 ; c vv = 1450 kj 1T 1 deg- 1 . 

* Op. cit., pp. 1772-1777. 

t c pv is converted to mechanical units from an estimated value given in Dorsey, 
op. cit., p. 599, line 4. The value of c vv , chosen so that our equation 3-07 (2) is satis- 
fied, is near the mean of two determinations reported by Dorsey, op. cit., p. 105. 



Section 3-07 46 

These values may be as much as 2% in error. With the chosen values, 
we have 

(2) c pv - c m = 461 R V1 

which agrees with 2-21 (4). Since water vapor is in some ways far from 
being a perfect gas, c pv - c vv may actually differ from R v , but the order 
of magnitude is correct. 

3 -08. Changes of phase. In sections 3-01 and 3-02 it was stated that 
at certain pressures and temperatures an equilibrium may exist between 
any two phases, for example, between liquid water and water vapor. To 
make the following discussion apply to all three phase-equilibria, we 
shall denote the phases by 1 and 2. The pressure at which the two 
phases are in equilibrium will be denoted by e 8 . This notation will be 
used throughout the chapter. 

Consider now a process where a unit of mass transforms at equilibrium 
from phase 1 to phase 2. The equation of energy 2-18(3) can be written 
in the following form, since dh = Tds by 2*28(1) : 

(1) dh = Tds = du+e a da. 



Here 5 is the specific entropy of the system. The complete change of 
phase will be represented by integrating (1) from phase 1 to phase 2; 



(2) 



Z Z Z Z 

I dh= I Tds= I du+ I e a da. 



The first integral represents the total amount of heat absorbed by the 
unit mass in phase 1 in order to cause it to transform completely into 
phase 2. It is known as the latent heat of the transformation 1 to 2, and 
will be denoted by Li 2 - Latent heat has the dimensions [L 2 T~~ 2 ] of 
specific energy. The pressure and temperature remain constant during 
the transformation (see fig. 3-01&), so that (2) may be integrated: 

(3) Li2 - T(s 2 - si) - (u 2 - ui) + e 8 (a 2 - e*i). 

The equation (3) will be used in later work. 

For the present it suffices to realize that, at each pressure e 8 and corre- 
sponding temperature T where two phases can co-exist, there is a definite 
latent heat Li 2 . 12 varies with temperature, and it is different for 
each of the three phase transformations. Also, Lj 2 is equal to the 
amount of heat released by a unit mass in phase 2 when it transforms to 
phase 1. (That is to say, Li 2 = -21-) 

The three possible phase transformations of water substance are: 
water - vapor (wv), ice<~> vapor (iv), and ice<-> water (iw). The corre- 



47 Section 3-09 

spending latent heats are called respectively: latent heat of evaporation 
(L wv , always written L), latent heat of sublimation (L t - v ), and latent heat 
of melting (Li w ). All these transformations will be discussed in detail 
later. 

For reference we give the values of the latent heats at 0C for water 
substance.* 

(4) L= 2.500 X 10 6 kj f 1 ; 

(5) I,*- 2.834 x It^kjIT 1 ; 

(6) L iw - 0.334 x 10 6 kj IT 1 . 

These values will be used for most purposes as constants in the atmo- 
spheric range of temperatures. 

3 -09. Variation of the latent heats with temperature. The variations 
of the latent heats with temperature are relatively small but can be ob- 
tained from theory. We shall give the argument for L. For the case 
of the latent heat of evaporation 3-08(3) takes the form 

(1) L = e s (a v - a w ) -f (u v - u w ). 

At atmospheric temperatures we may neglect a w (= 1) against a vt which 
was seen in 3-06(8) to exceed 25,000. By replacing e 8 a v by R V T from 
3-06(1), (1) becomes: 

L = R V T + u v - u w , 
or in differential form 

(2) dL = RjlT 4- du v - du w . 

Since the vapor behaves nearly like a perfect gas, we note from 2-20(7) 
that du v = c vv dT. Since a w is practically constant, the energy equation 
2-18(3) for the liquid reduces to dh = du w . And from the definition of 
c w , we see that dh = CwdT. Thus du w = c w dT, which when introduced 
in (2) yields 

dL = (R v 4- c vv - c w )dT = (c pv - c w )dT. 

The last step is according to 3-07 (2) . Finally, dividing by dT we get 

dL 

(3) = c pv - c w , 

al 

which contains the following simple rule: Tlte rate of change of the latent 
heat L with temperature is equal to the change of the specific heat at constant 
pressure from the liquid to the vapor phase. Note that (3) checks dimen- 
sionally. 

* Dorsey, op. cit., pp. 616-617. We assume L& = L 4* Li w . 



Section 3-09 48 

Integrating (3) and using the values of the specific heats in 3-05 and 
3-07, we have in the atmospheric range -40C to 40C the good approxi- 
mation : 

L - Lo + (7 - To) (c pv - c w ) - Lo - 2274(r - 273), 

where LQ is the value of L at 0C in 3-08(4). Introducing this value, we 
get the final formula 

(4) L = (2.500 -0.002274*C) X 10 6 kj f 1 (in the atmosphere). 

Inspection of (4) justifies the ordinary approximation that L is constant 
in the atmosphere. 
The reader may show similarly that 

(5) L iv - (2.834 - 0.000149/C) X 10 6 kj t" 1 (near 0C). 

Thus Li v has a variation with temperature only 7% as large as that of L. 
Hence L t - v is still more appropriately taken to be constant. 

For L{ W an essential modification of the above argument is required. 
Since, however, the melting temperature of ice is almost constant in the 
atmosphere (see 3- 13), the dependence of L iw on temperature is unim- 
portant to meteorology. On the other hand, (4) and (5) are sometimes 
used in meteorological investigations requiring accuracy. 

3 10. Clapeyron's equation. Consider water substance in two 
phases, called 1 and 2 as in 3-08. For each temperature T less than T c , 
there is one saturation pressure e 8 at which the phases 1 and 2 are in 
equilibrium. Conversely, for each pressure e less than e c , there is one 
transformation temperature at which the equilibrium exists. Let the 
two phases be numbered so that in equilibrium s 2 > $i i- e - so that heat 
must be added to phase 1 to convert it to phase 2. 

It is now our purpose to obtain a differential relationship between the 
saturation pressure e s and the transformation temperature T just defined. 
From the latter equality of 3-08(3), we have at the state (7> a ) : 

(1) ui 



where the subscripts refer to the two phases. Both sides of (1) have the 
same form, showing that the function 

(2) (f>= u + e 8 a- Ts 

remains constant during the iso thermal -isobaric change of phase. The 
function <p is known as the thermodynamic potential, and it is a function 
of state alone. It has the dimensions [L 2 T~ 2 ] of specific energy. 

We now consider the isothermal change of phase at the temperature 
T + dT and the corresponding pressure e 8 + de 8 . As above, the thermo- 



49 Section 3- 10 

dynamic potential will be constant throughout the change of phase at 
T + dr. Let its value be <p + d<p. By differentiating (2), we get 

dtp = du + e a da Tds + ade s - sdT. 

But the first three terms on the right are zero, according to the energy 
equation 3-08(1). Thus 

(3) d<p = ade 8 - sdT. 

Since ^? remains constant in the change of phase at (T,e 8 ), and since 
<p + d<p is constant in the change of phase at (7" + dT,e 9 -\- de 8 ), it follows 
that d<p remains constant during the transformation from the phase 1 to 
the phase 2. We have therefore from (3): 



or by rearrangement : 

de 8 



(4) 



dT Ot<2 Oil 



But $2-51 = 1,12/7", f rom 3-08(3). Hence we get the desired final 
differential relationship between e s and T: 

Ll2 



dT T(ct 2 OL\) 

Equation (5) is Clapeyron' s equation, found by Clapeyron in 1832 and 
later derived from the modern point of view by Clausius. 

Since Clapeyron 's equation holds for any two phases, we may write 
down the three forms it takes: 

de 8 L 

(6) = - - (water^ vapor) ; 
al 1 (a v a w ) 

ti\ ^ e * ^ iv r \ 

(7) 7^.= ^ - T (ice^-> vapor); 
dT T(a v - oti) 

/rt\ (*&& J-^IW ,. V 

(8) J^=~^( - N (ice<-> water). 
al l (a w - oti) 

For equilibrium between each pair of phases the pressure e 8 and tempera- 
ture T satisfy the corresponding Clapeyron equation. Each of the 
equations may be integrated to give a curve in the (T,e) -diagram, which 
we shall call respectively the evaporation curve, the sublimation curve, and 
the melting curve. Along the evaporation curve there exists equilibrium 
between water and vapor. Along the sublimation curve there exists 
equilibrium between ice and vapor. At the point where these two curves 



Section 3-10 



50 



intersect there is equilibrium among all three phases. It follows there- 
fore that the melting curve, representing equilibrium between ice and 
water, must pass through the intersection of the other two curves. This 
common point on all three curves is the triple point, and the correspond- 



Critical point- 



Ice 




Evaporation 

curve 

^^x^Triple point 
Sublimation curve 



Vapor 



m ^ 

FIG. 3-10. 

ing state is the triple state mentioned in 3-02. A schematic (7-dia- 
gram is given in fig. 3-10. 

In the next sections we shall give a more detailed discussion of each 
of the three changes of phase of water substance. 

3-11. Saturation vapor pressure over water. In the case of the 
water-vapor transformation, the pressure e 8 is called the saturation 
vapor pressure (over water). The corresponding temperature T at which 
the transformation takes place is called the evaporation temperature (or, 
sometimes, boiling temperature) . The curve showing the variation of e s 
with T on a (!T,e)-diagram is the evaporation curve, and 3-10(6) is the 
differential equation of this curve. The evaporation curve is known to 
pass through these three points: 

(triple point) ; 
(normal boiling point) ; 
(critical point). 

The critical point is of course one end of the evaporation curve. 

Now a w = 1 and a v > 25,000 in the atmosphere (see 3-06). Using 
the equation of state 3-06(5), we can therefore replace a v a w by 
a v - RdT/ (ee 8 ), with sufficient accuracy. Then 3-10(6) takes the form 



(1) 


e g = 0.611 cb, 


T - 273K 


(2) 


e, - 101.33 cb, 


T - 373K 


(3) 


e 8 = 22,100cb, 


T = 647K 



(4) 



J. de. 

e,dT" 



(water *- vapor), 



51 



Section 3-11 



an important form of Clapeyron's equation. Equation (4) can also be 
written 

(5) d(lne 8 )= ~lT d [^ 

Rd V, 

In 3-09 it was shown that in a limited range of temperatures (such as 
the atmospheric range) L may be regarded as constant. With this 
assumption (5) represents a straight line on a (- 1/ T, In e) -diagram. 
By using the value L = 2.500 x*10 6 kj t" 1 , (5) becomes 

(6) d(ln e 8 ) = -5418 d ( - j (water <-> vapor, -40C to 40C). 
Or in base 10 logarithms: 

(7) d(loge s )= -2353d(-j (water <--> vapor, -40C to 40C). 

Integrating (7), with the initial condition (1), gives 
2353 



(8) log e, - 8.4051 - 



(water** vapor, -40C to 40C). 



Equation (8) provides a convenient graphical method of determining e a 
in the atmospheric range. Fig. 3-1 la shows a graph of log e against 
1/r, for temperatures up to the critical. In the atmospheric range, 
the slope of the evaporation curve is 2353, which is represented by the 



10 5 
10 4 

tlO 3 

5 100 

10 

1 

0.1 cb 




Critical point 

C 
B 

Normal 
boiling point 



Triple point 



250 K 350 450 650 



FIG. 3-1 la. 




Ocb 



line AB in the figure. The variation of L at high temperatures causes 
the true evaporation curve to bend slightly (curve AC), so as to go to 
the critical point. The evaporation curve may then be transferred to a 
linear (T,e) -diagram, where the curve shows extreme curvature (see 
fig. 3-116). All physical and meteorological handbooks contain tables 
giving e 8 as a function of 7", as determined from experiments. 



Section 3- 11 52 

If the formula 3-09(4) for L is introduced into 3*11(4), an integration 
can be carried out to yield the Magnus formula for e 8 : 

2937 4 
(9) log e s - - - 4.9283 log T+ 22.5518 

(water*--* vapor, -40C to 40C). 

Table 3-11 shows in parallel rows the values of e s from the simple formula 
(8), from (9), and from empirical data. (The observed values at tem- 
peratures below 20C are unreliable, since the water tends to freeze.) 
As might be expected, the values derived from (9) show closer agreement 

TABLE 3-11 
T-TQ -30C -20C -10C 0C 10C 20C 30C 

e a from (8), cb 0.0527 0.1273 0.2873 0.611 1.232 2.37 4.36 
^from(9) 0.0509 0.1254 0.2862 0.611 1.228 2.339 4.247 

^observed ............ 0.2865 0.611 1.228 2.338 4.243 

with the observed values than those derived from (8), but the difference 
is less than one millibar throughout the atmospheric range, and for the 
most common temperatures the simple formula (8) gives all the accuracy 
needed. 

312. Saturation vapor pressure over ice. The ice<~> vapor trans- 
formation can be treated in complete analogy to 3-11. Since the re- 
semblance is so close, and since the transformation is less important, 
we shall go over it rapidly. 

In the case of the ice<~>vapor transformation, the pressure e s is called 
the saturation vapor pressure over ice. The transformation temperature 
T is called the sublimation temperature at the pressure e 8 . The curve 
giving e s as a function of 7" is the sublimation curve. As stated in 3*10, 
the sublimation curve is known to pass through the triple point of 
3-11(1). 

Clapeyron's equation 3-10(7) is the differential equation of the subli- 
mation curve. Repeating the argument which led to 3-11(4), we may 
obtain the differential equation in the form 

/(gx 1 de s eLtv 

(1) - = -2 (ice <--> vapor). 

e s al 



In the atmospheric range L iv may be regarded as constant. Taking its 
value from 3-08(5), we get the analog of 3-11 (8) : 

^fifii 
(2) log e s = 9.5553 - =- (ice~> vapor). 



53 



Section 3- 13 



The sublimation curve plotted on a graph of log e against -1/T is a 
straight line (fig. 3-12a). For comparison the evaporation curve is 
drawn as a broken line. The sublimation curve is steeper than the 
evaporation curve and lies at slightly lower pressures. It should be 



Icb 



0.5 



B al 

0.05 



Triple point 




0.8cb 



0.6 



0.4 



0.2 



230 K 250 270 



FIG. 3-12o. 



O.O 



Triple point 




230 K 250 270 



FIG. 3-126. 



290 



noted that the sublimation curve does not extend above the triple point, 
since ice cannot be heated above this temperature. However, water 
can be cooled below its freezing point, and such supercooled water is 
usually found in the atmosphere. (See 345.) Consequently the 
evaporation curve must be drawn to about -30C. 

In fig. 3-126 the same curves are drawn on a linear (T,e) -diagram. 

3'13. Pressure and temperature of melting. The treatment of the 
ice->water transformation is similar to that of 341 and 3-12, but it 
differs in that the equation of state for water vapor does not enter into it. 
In the case of the ice-water transformation, the pressure e s is called the 
melting pressure of ice, and the corresponding temperature T is called 
the melting temperature of ice. Whenever ice and water are in equilib- 
rium with each other at a surface of mutual contact, the hydrostatic 
pressure of the water on the surface is the melting pressure and the 
temperature is the corresponding melting temperature. 

The curve giving e s as a function of T is the melting curve. It passes 
through the triple point 

(1) e 8 = 0.611 cb, T - T Q = 0.0075C, 
and also through the normal melting point 

(2) e 8 - 101.33 cb, T - T Q - 0C. 



Sect ion 3- 13 54 

The differential equation of the melting curve is Clapeyron's equation 
in the form 3-10(8): 

/. N 

(ice<-> water). 



al 1 (a w oii) 

Introducing the practically constant values: LI W = 0.334 x 10 6 from 
3-08(6), ai = 1.091 from 3-04(1), and a w = 1.00 from 3-05(1), we get at 
T= 273K: 

de 

(3) - - 13,440 cbdeg"" 1 (ice<-> water). 

al 

We note from (3) that the melting curve is the only one of the three 
phase-transformation curves which has a negative slope in the (7- 
diagram ; i.e., the melting temperature decreases with increasing pressure. 
We note also that the curve is very steep, being nearly isothermal (see 
fig. 3-10). 

As a verification of (3), we may compute the theoretical pressure differ- 
ence between the normal melting point (1) and the triple point (2). 
Introducing dT = 0.0075 in (3), we get de s = 100.8. The empirical 
value is 100.7 cb, from (1) and (2). 

The drop in melting point with increasing pressure has the following 
effect. When ice at a temperature slightly below 0C is subjected to 
high pressure, it is brought into a state above its melting point. It is 
therefore converted into water but freezes again as soon as the pressure 
is released. This phenomenon is called regelation, and it accounts for the 
plasticity of ice which permits the flow of glaciers. 

3-14. Complete (J,e)-diagram. In fig. 3-10 are combined into one 
schematic diagram the three transformation curves just discussed. For- 
getting about the dotted extension of the evaporation curve below the 
triple point, we see that the (I-diagram is divided into three regions. 
These represent the ice, water, and vapor phases; they are labeled 
accordingly. The phase of water substance is uniquely determined by 
its temperature T and pressure e, except along the curves where two 
phases can exist in equilibrium. 

Any equilibrium process must have a continuous path on the (T,e)~ 
diagram. Thus a change of phase must occur on one of the transforma- 
tion curves and ordinarily will occur only on these curves. An excep- 
tion is found when we pass from the vapor region to the water region 
through pressures higher than the critical. This corresponds to a con- 
tinuous physical process with none of the usual properties of a change of 



55 Section 3-16 

phase. There is probably likewise a critical point on the melting curve 
with similar properties. There is, however, no critical point on the 
sublimation curve, which must extend to absolute zero. 

3-15. Supercooled water. The dotted extension below 0C of the 
evaporation curve has a physical significance. Suppose that we have 
vapor and water in equilibrium, and cool them carefully. At the triple 
point the water ordinarily freezes but, if the water is pure, the ice phase 
may fail to appear. The vapor-water combination continues to cool and 
follows the evaporation curve. This supercooled state is thermodynami- 
cally unstable, and the slightest disturbance will make the system jump 
into the stable state on the sublimation curve. 

The droplets in clouds and fogs which are formed by condensation 
above 0C will usually assume the supercooled liquid state on cooling 
below 0C. According to experience most cloud elements are still 
liquid at 10C, and water droplets may be found down to -30C. 
When ice particles are brought into a cloud of supercooled drops at a 
fixed temperature, the system is no longer in equilibrium. The vapor is 
saturated with respect to the water drops, but is supersaturated with 
respect to the ice particles; i.e., e is larger than the saturation vapor pres- 
sure over ice (see fig. 3-12&). The result is condensation of vapor on the 
ice particles. But the loss of vapor means that e becomes less than the 
saturation vapor pressure over water. Thus water evaporates. The net 
result of the two processes is a growth of the ice crystals at the expense 
of the water droplets. This goes on until all the water drops have 
evaporated. This process goes on most rapidly near 12C, where 
the saturation vapor pressure over water most greatly exceeds that over 
ice. 

Two important meteorological applications of the above phenomena 
may be mentioned: (i) When a supercooled fog moves over a snow- 
covered surface, it tends to dissolve; (ii) when a relatively small number 
of ice crystals are present in a cloud of supercooled water droplets, the 
ice crystals grow enormously. They can therefore no longer remain 
suspended in the air, and they start falling. Bergeron assumes that 
tiny ice crystals are always present above a certain level as the end 
product of disvsipated cirrus clouds. When a cloud grows in thickness 
precipitation may be expected from it when its top has reached the ice 
crystal level. Although the indications are that other factors also con- 
tribute to the formation of precipitation, the effect mentioned here is 
undoubtedly an important one. 

3*16. Thermodynamic surface of water substance. The fact that 
there are three variables of state, namely, e, T, and a, suggests a three- 



Section 3-16 



56 



dimensional representation of the state of water substance. Let three 
coordinate axes measure e, T, and a, respectively. Each state (e,T,a) of 
water substance is then represented by a unique point in space. 

As shown in 2-07, water substance cannot assume an arbitrary state 
(e,T,a), but only those states (e^T^ot) which satisfy an equation of state 
of type 

(1) f(e,T,a) = 0. 

It is not possible to express relation (1) exactly in terms of one elemen- 
tary function. The equation of state can, however, be expressed approxi- 
mately in a restricted range of states. See 3'06(5), for example. 

The nature of the equation of state is such that all the points (e,T,a) 
satisfying (1) lie on a continuous surface. This is called the thermo- 
dynamic surface for water substance and is a representation of all possi- 
ble states of water substance. This surface is shown in fig. 3-16. Each 



Ice and water 
(hidden) 




FIG. 3-16. Thermodynamic surface of water substance. 

phase is represented by an area on the surface. Each region of equilib- 
rium between two phases is represented on the surface by an area where 
the isotherms are parallel to the a axis. The thermodynamic surface 
reduces to fig. 3-02 when projected on the (a,e)-plane, i.e., when 
viewed parallel to the T axis. It reduces to fig. 3*10 when projected 
on the (7-plane, i.e., when viewed parallel to the a axis. 

The construction and study of a model of the thermodynamic surface 
will greatly enhance the understanding of the thermodynamic properties 
of water substance. 



57 Section 3-18 

317. Moist air. Thus far in chapter 3 we have treated the thermo- 
dynamics of pure water substance. The real atmospheric air with its 
variable admixture of water vapor will be called moist air. Its thermal 
properties are obtained by combining the thermal properties of the dry 
air and the water vapor. The question arises whether the presence of 
the dry air constituents in any way influences the thermal behavior of 
the water vapor. As long as the vapor is unsaturated it behaves very 
closely as a perfect gas and, according to Dalton's law (see section 2-12), 
its state is unaffected by the presence of the dry air. When moist air is 
brought in contact with a water surface, equilibrium is reached when 
there is equilibrium between the water vapor in the air and the liquid 
water. This situation is of course not identical with the one discussed 
in section 341 where no foreign substance was present. We have no 
right to assume a priori that the saturation vapor pressure will be the 
same in the two cases. However, it has been found that for practical 
purposes the atmosphere does not influence the saturation vapor pressure; 
i.e., the partial pressure of water vapor in saturated air is equal to the 
saturation pressure of pure water vapor. 

3-18. Moisture variables. Dry air is treated as an invariable per- 
fect gas, according to section 2*13. Since, however, the proportion of 
water vapor in the atmosphere varies greatly, we must introduce vari- 
ables measuring the moisture content of a parcel of air. 

The first of these is the (partial) vapor pressure e of water vapor in the 
parcel. Here e has the same meaning as heretofore in chapter 3. It 
has the dimensions fML~ 1 T~~ 2 ] of pressure. 

Two other moisture variables arc the dimensionless ratios w and q 
now to be defined. Let the parcel of moist air with total mass M con- 
sist of Ma tons of dry air mixed with M v tons of water vapor. Thus 
Md +M V = M. Then the mixing ratio w is defined by: 

(1) w=* - (pure number). 

Md 

Thus w is the mass of water vapor per ton of dry air. 
A closely related variable is the specific humidity q defined by: 

M 

(2) q = -77 (pure number). 

M 

Thus q is the mass of water vapor per ton of moist air. 
Now we see that 

1 M M v +M d _ 1 Md =l I, 
q ~~ M v M v M v w 



Section 3- 18 58 

whence by solving for q and w: 

w q 

(3) q = and w = - 

1 + w 1 - q 

For absolutely dry air, w = q = 0. For pure water vapor, w = oo and 
q = 1. q is always less than w. It will be shown in 3-20 that in the 
atmosphere usually w < 0.04. Then from (3) we see that q/w > (1.04)" 1 
0.96. Thus q and w differ by at most 4%, and usually much less. 
For most practical purposes, we can use w = q. For this reason the names 
of w and q have been confused in meteorological literature. Logically 
one of the two variables could well be omitted, but current usage makes 
it necessary to know both. 

The numerical values of w and q are found between and 0.04. We 
shall usually express these numbers as parts per thousand. For exam- 
ple, the value 0.0154 will be written 15.4 x 10~ 3 . Some authors omit 
the factor 10~ 3 , and speak of " 15.4 per mille " (15.4 %o) or " 15.4 
grams per kilogram.' 1 In applying these numbers to thermodynamical 
formulas, the factor 10~~ 3 must be added. 

3-19. Relative humidity. The maximum vapor pressure obtainable 
at a given temperature is the saturation vapor pressure over water (not 
over ice). This will be denoted by e s throughout this discussion. In 
3'11 it was shown that e s depends only on the temperature, and in 3'17 
we mentioned that the presence of dry air leaves this property of e 8 
unchanged. Any attempt to raise e above e s will usually cause conden- 
sation to liquid water. 

The values of w and q for saturated air are called the saturation mixing 
ratio (w s ) and the saturation specific humidity (q s ) respectively. Thus 
we have 

(1) e ^ e 8 \ w ^ w a ] q^ q 8 . 

Many meteorological and physiological phenomena involving the 
" wetness " of the air depend not on the quantity (w) of vapor present 
but rather on the degree of saturation. This is true of the hair hygrome- 
ter used in meteorographs to measure vapor content. It is also true of 
the comfort of a person living in the air. The degree of saturation could 
be measured by any of the quantities w/w a , q/q 8 , or e/e s . It is the uni- 
versal practice to use the third, called the relative humidity (r) . We have 

(2) r=^- 

e 8 

Thus r lies between and 1.00. In practice it is expressed as a per cent. 
If r = 22%, then r is to be given the value 0.22 in our formulas. In 



59 Section 3-20 

words, f indicates what per cent the actual vapor pressure is of the- 
saturation vapor pressure over water at the same temperature. 

The meteorograph reports relative humidity as a primary measure- 
ment. Thus the state of the upper air is originally described by total 
pressure p, temperature T, and relative humidity r. The main problem 
of the next sections is to obtain the values of w, q, and e from p, T, and r. 
This will be done both numerically and graphically. 

3 '20. Relations among the humidity variables. The first task is to 
express w in terms of e. The total pressure p of a parcel of air is by 
Dal ton's law (242) the sum of the partial pressure pd of the dry air and 
the vapor pressure e of the water vapor. Thus 

(1) Pd=p-e. 

Let Md tons of dry air and M v tons of vapor separately fill an entire 
volume V. We write the equations of state of each component sepa- 
rately in the form 2-11(1): 

eV = MvR v T (water vapor); 
p d V= M d R d T (dry air). 

We take the ratio of these equations, and introduce M v /Md = w from 
3-18(1) and R v /R d = 1/e from 3-06(3). We then have 

(2) ^ W ' 

Pd 6 

Solving for w and using (1), we get 

(3) w = (exact). 

p- e 

Thus w is obtained from p and e. This is a very important formula to 
know. If the air is saturated, we get as a special case of (3) : 



(4) W '=T~ ~ (exact). 

p- e s 

We can now estimate the largest value of w s likely to occur in the 
atmosphere, say that for saturated air at 36C and 100 cb. From 
3-11 (8) we determine that e s 6 cb. Then from (4) 



In everyday synoptic work, w rarely exceeds 20 X 10~~ 3 . 
Solving (3) and (4) for the vapor pressure, we get the occasionally 



Section 3-20 60 

useful relations: 

(5) --3L; ,.-*L (exact). 

c + w e+w 8 

The relations (5) can be replaced for most practical purposes by the 
important approximate formulas 

P P 

(6) e ~ - w\ e a w 8 (approximate). 



Formulas (6) are obtained by ignoring w and w s in the denominators of 
(5), since these are quite small in comparison with c. Formulas (6) 
are the more accurate, the smaller w and w s are. The error in (6) rarely 
exceeds 3%. 

3*21. Numerical determination of mixing ratio. Now suppose p, T, 
and r are known, and w is desired. The following calculation is carried 
out: (i) The saturation vapor pressure e s is obtained from T alone, 
according to formula 3-11(8); (ii) we get e = re s , from 3-19(2); (iii) we 
get w from p and 6, by 3-20(3). If desired, q can be obtained from 
348(3), or with a small error q = w. Thus the problem stated above is 
solved in principle. This should be carefully understood. 

However, this algebraic procedure involves laborious numerical com- 
putations for each determination of w. For all the accuracy required in 
practice, a graphical procedure is much quicker. This will be described 
in section 3-23. 

Another important problem is one that will arise in the use of the dia- 
gram : given w s and r, to determine w. The exact solution is independent 
of p and could be obtained as follows: (i) Get e s from 3-20(5); (ii) then 
get'*e from 3-19(2); (iii) then get w from 3-20(3). The result of this 
algebra would be 



The following approximate solution is the one invariably used in prac- 
tice: Take the ratio of the two equations 3-20(6), obtaining 



w 



Hence 

(2) w rw 8 (approximate). 

A comparison of (2) and (1) shows that (2) is exact when r = and also 
when r - 100%. It can be shown that w obtained from (2) is usually 
in error hy not more than 0.2 x 1(T 3 , which is accurate enough for most 



61 Section 3-23 

purposes. The error can be as much as 0.6 x 10~ 3 , when w a = 40.0 x 
10~ 3 and r - 50%. 

3 -22. Vapor lines on the diagrams. Since e s is a function of T alone 
(see 341), it follows from 3-20(4) that w 8 is a function of p and T. Since 
all the meteorological diagrams contain the variables p and T, it is 
possible to draw curves of constant w s on each diagram. These curves 
should properly be called mixing ratio lines for saturated air, but we shall 
refer to them simply as vapor lines. 

The shape of the vapor lines depends on the diagram. The differen- 
tial equation of a vapor line may be obtained as follows: From 3*20(5) 
if w 8 const, then e s = (const) p. Hence d(log p) = d(log e 8 ). Intro- 
ducing the expression 3-11(7) for (/(log e s ), we have 



(1) d(\og p) = -2353 d ( - j (vapor line). 

By integrating (1) it is seen that on the emagram each vapor line is a 
segment of a hyperbola, and that all vapor lines are congruent. 

In the atmospheric range, the vapor lines are nearly straight lines on all 
diagrams. They always have a slope between that of the dry adiabats 
and that of the isotherms. That is, on each vapor line as p decreases, 
T decreases but 6 increases. The vapor lines on a tephigram are shown 
in fig. 3-36. The student should study the vapor lines on all the dia- 
grams at his disposal. See 3-18 for the usual method of labeling them. 

323. Graphical determination of w, <f, and e. We now consider the 
graphical solution of the problem mentioned at the end of 3-19: given 
p, T, and r, to determine w, q, and e. 

(i) To determine w, first plot the point (T,p) on any thermodynamic 
diagram. Interpolating between the vapor lines, find the value of w 8 \ 
this should be accurate to 0.1 X 10~ 3 . Finally, multiply w 8 by r to 
obtain w, according to 321(2). 

(ii) To determine q, simply take q = w. More accuracy is neither 
necessary nor compatible with the approximation already made in (i). 

(iii) To determine e, first obtain the value of e 8 . Since e s depends on 
temperature alone, it will be the same at the point (T, 62.2 cb) as it is 
at the point (T,p). Read the value w 8 of the saturation mixing ratio at 
the point (T, 62.2 cb). Then by 3-20(6), 

1 



Hence e 8 10 3 ze^ millibars. Finally, the value of e is equal to re B . This 
completes the graphical solution of the problem mentioned. 



Section 3-23 62 

The rule for getting e a at any temperature can be expressed in words: 
The value in millibars of the saturation vapor pressure e a at the temperature 
T is approximately equal to the value in parts per thousand of the satura- 
tion mixing ratio at the temperature T and pressure 622 mb. Thus a dia- 
gram can replace a table of vapor pressures, with an error rarely exceed- 
ing 4%. 

One additional step will correct practically all this inaccuracy in e 8 . 
Having found iv' a above, express it in parts per thousand, and add it to 
622, to get a pressure pi in millibars. Then go to the point (T,pi) 
and read the value w' f of the saturation mixing ratio. This will be 
almost exactly e 8 in millibars. The reason follows from the exact for- 
mula 3-20(5): 

Pi tt Pi n 62.2 -MOV' ft 1 

.- ^ % + w ^ - - 



The error committed here is negligible, compared with that in (1 ). 

3*24. Thermal properties of moist air. Moist air for which the rela- 
tive humidity is 100%, i.e., e= e a , is called saturated. Otherwise the 
moist air is called unsaturated (e < e s ). We have seen that the water 
vapor seldom comprises more than a few per cent of the air. As a result, 
during any process which does not lead to condensation, moist air 
behaves like a perfect gas whose thermal properties differ only slightly 
from those of dry air. 

In 3-25 and 3-26 will be given the equation of state and the values of 
the specific heats for moist air, using the general theory of chapter 2. 
There will be given explicitly the small deviations from dry air caused 
by the presence of vapor. These deviations will be expressed in terms 
of the parameters w and q expressing the variable vapor content of moist 
air. 

In section 3-27 the adiabatic process for unsaturated air is considered. 
This also is nearly the same as that for dry air. However, as soon as 
saturation is reached in the adiabatic process, any further cooling leads 
to condensation with consequent release of relatively large amounts of 
latent heat. As a result there is a sudden transition to new types of 
adiabatic processes, to be discussed in sections beginning with 3-30. 
Thus it is vitally important to distinguish between the behavior of air 
in its so-called unsaturated stage and its behavior in the saturated stage. 
The unsaturated stage is called by some authors the dry stage, although 
it does not deal with truly dry air. 

The notation used here is in so far as possible governed by the follow- 
ing principles. For dry air the constants and variables are given the 



63 Section 3-25 

subscript d. For water vapor they are given the subscript v. For moist 
air and hence for the atmosphere in general they are given no subscript. 
However, the adoption of a notation is always limited by the usage well 
established by previous writers. There are also difficulties inherent in 
the nature of a science. As a result, there will always be certain incon- 
sistencies. For example, the vapor pressure might well be denoted by 
p vj but e is always used. The temperature T is the same for all compo- 
nents of a system in equilibrium; hence no symbols T d or jT v are needed. 

3*25. Equation of state of moist air; virtual temperature. Let a 
parcel of one ton of moist air have the specific humidity q. Then the 
parcel contains Md = 1 - <z tons of dry air and M v = q tons of water 
vapor. According to 242 (3), the specific gas constant R of the mixture 
is given by 

1-R-MdRd+MJR*, or 

( } 



But by 3-06 (3) we can write R v = R d /e. Hence 



Evaluating (1/e) - 1 and using q w t we get the important formulas 
(2) = (1 + Q.6lq)Rd (1 + Q.6lw)R d . 

Thus the presence of water vapor will raise the specific gas constant of 
atmospheric air from the value Rd = 287 up to a maximum of 294 (cor- 
responding to q - 40 x 10~ 3 ). 

Using (2), we can write the equation of state of moist air: 



(3) p a =RT= R d (l + Q.6lq)T. 

We see from (3) that moist air of specific humidity q in the state (p,T) 
has the same specific volume as dry air would have in the state (/>,jT*), 
where 

(4) r* - (1 + 0.61g)r (1 + 0.6lT;)r. 

The temperature T* is called the virtual temperature of the moist air and 
is by definition the temperature of dry air having the same pressure and 
specific volume as the moist air. 

The expressions (3) and (4), when combined, give an alternative form 
of the equation of state for moist air : 

(5) ' pa - 



Section 3-25 64 

Thus to compute a. we have our choice of using the real temperature T 
with an altered gas constant -R, or of using a fictitious temperature T* 
with the dry air gas constant R*. The former method of equation (3) 
is perhaps simpler for computations. The use of virtual temperature and 
(5) affords a great simplification of later dynamical theory. 

3 26. Specific heats of moist air. Consider the one- ton parcel of 
moist air described in 3-25 above. Let there be introduced the quantity 
dh of heat into the parcel. As a result the parcel is heated from tempera- 
ture T to T + dT. This temperature rise dT is experienced by both the 
dry air and the water vapor. Let dh d be the amount of heat received 
by the dry air , per ton of dry air. Let dh v be the amount of heat received 
by the water vapor, per ton of water vapor. Then in the notation at the 
start of 3-25: 

(1) dh= M d dh d + 



By dividing both sides of (1 ) by dT and writing Md and M v in terms of g, 
we have 

dh dh d dh v 

(2) = (1 -<Z) -7-4-27,7; ' 
dT di dT 

Now (2) holds for an arbitrary process in which q (and hence w ) are 
constant. If the process is at constant pressure p for the whole parcel, 
then from 3-20(5) we find that it proceeds at a constant partial pres- 
sure e for the water vapor and hence at a constant partial pressure 
p d = p - e for the dry air. At constant />, dh/dT becomes by 2-19(1) 
the specific heat c p of moist air at constant pressure. Since e and pa are 
constant, the other two quotients in (2) are specific heats at constant 
pressure. Hence from (2) we have 

c p - (1 ~ <l)Cpd + q c pv c pd \ 1 + (^ - M q 

Evaluating c pv /c p d from 3-07(1) and 2-21(5), we have the important 
formulas 

(3) c p = (1 + 0.90q)cpd (1 + 0.9Qw)c pd . 

Thus the presence of water vapor will raise the specific heat of air from 
c p d = 1004 to a maximum of 1040. A similar analysis for the specific 
heat c v of moist air at constant volume gives the less used formulas 

(4) c v - (1 -f l.Q2q)c vd (1 + l.Q2w)c vd . 

3 -27. Adiabatic process of unsaturated air. The adiabatic process of 
unsaturated air (the unsaturated stage of 3-24) is a special case of the 



65 Section 3-27 

adiabatic process of any perfect gas. A certain parcel of moist air is 
under consideration. Since no condensation takes place in the unsatu- 
rated stage, the value of q is constant. Then R, c v , and c p have the 
numerical values given by 3-25(2), 3-26(3), and 3-26(4). With these 
values, the equations of section 2-24 define the adiabatic process exactly. 
There only remains in theory to evaluate the constants K = R/c p and 
t\ = c p /c v of 2-24. We have 

1 + 0.61? . , 



(1 + 0.90tf)c p< i 1 + 0.90(2 

where K# = 0.286 by 2-24(12). Since q is always small, we can expand 
(1 -f O^Og)" 1 in a power series and then can neglect squares and higher 
powers of q. We finally obtain with all necessary accuracy the impor- 
tant formula 

(1) K (1 - 0.29q)K d (1 - Q.29w) Kd . 

A similar derivation gives a formula important in acoustics, but not 
used by us: 



We see from (1) that the presence of water vapor will lower Poisson's 
constant K from its dry air value of 0.286 to a minimum of 0.283. 

Let a parcel of moist air be in the initial state (T,p) , where p < 100 cb. 
According to 2-24(10), the temperature T\ and pressure p\ at any other 
state in the unsaturated adiabatic process will be given by: 

(2) T, = 

The equation (2) defines a curve which might be plotted on any meteoro- 
logical diagram, to be called an unsaturated adiabat. Since K varies with 
q, through (T,p) there would be a different unsaturated adiabat for each 
value of q. Let q be fixed, and consider the corresponding adiabat (2). 
This curve will intersect the 100-cb isobar at a temperature U , where 

(3) U = 

Now on the diagram there is a dry adiabat through (7\/>) with the poten- 
tial temperature 6, where by 2-26(1): 

(4) 8-: 

This 6 is the temperature which dry air in the state (T,p) would attain 
after adiabatic warming to 100 cb. Comparing (3) with (4) and remem- 



Section 3-27 66 

bering that K is slightly less than *d, we see that 6 U is slightly less than 0. 
Thus the adiabatic change of temperature with pressure is slightly less for 
moist air than for dry air. This rule holds for adiabatic cooling in the 
unsaturated stage, as well as for warming. It is ultimately a conse- 
quence of the water vapor's greater heat capacity. 

From (3) and (4), it can be shown that the dry adiabat and the steep- 
est unsaturated adiabat through the same point at 40 cb will have a 
temperature difference of about one degree at 100 cb. Hence in practical 
problems the dry adiabats can safely be used for moist unsaturated air with- 
out committing a significant error. This is invariably done in practice, 
and the unsaturated adiabats are never drawn on a diagram. 

The potential temperature 6 has so far been defined only for dry air. 
For moist air in the state (T,p) we have the choice between defining 0: 

(1) by (4) above; or (ii) as equal to the B u of (3) above. Definition (i) 
permits to be read directly from the dry adiabats of the meteorological 
diagram. Definition (ii) would preserve the property that 6 is equal to 
the temperature of the parcel after adiabatic compression to 100 cb. 
No definition can do both of these things. 

In order to keep our theory in the closest harmony with meteorological 
practice, definition (i) has been chosen, so that for moist or dry air, 
potential temperature is defined by the equation (4) above, i.e., by 2-26(1). 
For practical purposes we shall assume that the moist unsaturated adia- 
batic process is represented by the dry adiabats = const. 

3*28. Virtual potential temperature; characteristic point. Accord- 
ing to the formula 3-27(4), every point (T,p) on a meteorological dia- 
gram has a definite value of 6 assigned to it, which may be read directly 
from a diagram by interpolation between the dry adiabats. It is some- 
times convenient to use the so-called virtual potential temperature 6*. 
This 6* is the value of 6 for the point (T*,p). According to 3-27(4), 

" 

For moist air in the state (T t p), T* is determined from 3-25(4), and 
from 3-27 (4). Introducing these into (1), we get 

(2) *-(! + 0.619)0 (1 + 0.61^)0. 

Thus 6* bears the same relation to that T* bears to T. 

Consider now unsaturated air in the initial state (T,p) and having the 
mixing ratio w. The point (T,p) plotted on a diagram is called the 
image point of the air. When this air performs an adiabatic process its 
image point moves nearly (see 3-27) along the dry adiabat through the 



67 Section 3-30 

point (T,p). For an adiabatic compression the air will always remain 
unsaturated, and its image point continues along the dry adiabat. 
When the process is an adiabatic expansion the image point will move 
along the adiabat to the point (T 8j p 8 ) where saturation is reached. 
(See fig. 3*38). Rossby has named this the characteristic point, and it is 
conveniently determined as the point where the dry adiabat through 
the image point intersects the vapor line w a w. Thus the coordinates 
of the characteristic point may be thought of as (0,w)> where 6 and w are 
the values of these respective variables at the image point. Note that 
6 and w remain unchanged throughout the adiabatic process from image 
point to characteristic point. 

The point (T 8 ,p 8 ) marks the end of the unsaturated stage. Further 
adiabatic expansion results in condensation of part of the water vapor 
with the release of latent heat which is added to the air. This requires a 
separate investigation, to be given in the sections beginning with 3-30. 

3*29. Useful approximate formulas. If a meteorologist has frequent 
occasion to use the virtual temperature T* or the moist air constants R, 
c p1 and K, the following formulas are recommended. The formula (1) 
for the virtual temperature is especially useful in working on a diagram. 

In using formula 3-25(4) in the lower atmosphere we can use the 
average value 273K for T. Thus we have 

(o.ei)(273)w 



This gives the formula 

(1) r* 

Formula (1) is almost exact at T = 273K, and is good enough for most 
purposes in the atmosphere. 

ForU,wehaveby3-2S(2)thatlZ 1^+0.611?^- 287 + 0.175 
For many purposes we can use 

(2) U 

For c p we have similarly from 326(3), 

(3) c p 1 

For ic we get from 3-27(1) a similar, but less accurate, formula 



All these formulas can be used mentally. 

3*30. The adiabatic processes of saturated air. When saturated air 
expands adiabatically it will continue to remain saturated during the 



Section 3-30 68 

process, and some of the vapor will condense to water or ice. If the air 
were enclosed in an adiabatic container during the expansion, the prod- 
ucts of condensation would remain in the system and would evaporate 
again if the process were reversed into an adiabatic compression. This 
process which is both reversible and adiabatic is called the reversible satura- 
tion-adiabatic process. 

In the atmosphere the above conditions are not usually satisfied. 
Condensation is in most cases followed by precipitation, so that some of 
the condensed water or ice is removed from the system. The extreme 
case where all the products of condensation fall out of the air is called the 
pseudo-adiabatic process. This process is evidently not reversible, and 
neither is it strictly adiabatic, since the condensation products remove 
some heat from the system when they fall out. (In fact, the system 
itself is constantly changing in mass and composition.) The real atmos- 
pheric processes lie somewhere between the two extremes just described. 

The reversible saturation-adiabatic process is divided into three stages: 
(i) At temperatures above 0C there is the rain stage, where the vapor 
condenses to water, and the water vapor has the saturation vapor pres- 
sure over water; (ii) at 0C there is the hail stage, where the condensed 
water freezes to ice; (iii) at temperatures below 0C there is the snow 
stage, where the vapor condenses directly to ice, and the water vapor has 
the saturation vapor pressure over ice. The pseudo-adiabatic process 
has only a rain stage and a snow stage, since no water is retained to be 
frozen in a hail stage. 

In the real atmospheric process, the temperature of the transition 
between the rain and snow stages is usually below 0C, owing to the 
tendency of the water droplets to remain in the supercooled liquid stage. 
The snow stage differs relatively little from the rain stage. The hail and 
snow stages would require a separate discussion, but are meteorologically 
less important than the rain stage. The effect of an unknown amount of 
mixing makes the true atmospheric processes differ somewhat from the 
ideal processes described above. For all these reasons we shall confine 
our discussion to the rain stage and make the approximation that the 
entire adiabatic process is in the rain stage. 

The adiabatic expansion of saturated air was first investigated by 
Hann, and by Guldberg and Mohn. A more nearly complete discussion 
was given by Hertz (1884) and Neuhoff (1901). The distinction 
between the reversible saturation-adiabatic and the pseudo-adiabatic 
process was made by von Bezold (1888). A thorough discussion of both 
processes with, numerical comparisons between them was made by 
Fjeldstad (1925). 

Both the reversible saturation-adiabatic and the pseudo-adiabatic 



69 



Section 3-31 



processes can be expressed in the form of differential equations involving 
d'F, dp, and dw 8 . By w a we mean the saturation mixing ratio over water 
in the state (T,p). Since the air is always saturated, w 8 is also the actual 
mixing ratio of the air. Immediately upon expansion from the charac- 
teristic point, the temperature starts to decrease. This causes w a to 
decrease. Hence some water vapor condenses to liquid water, releasing 
some latent heat. This heat is used to warm the whole system, includ- 
ing the moist air. Hence the cooling proceeds at a slower rate than in the 
dry-adiabatic process. As a result of this argument, it follows that the 
adiabats for saturated air must have a slope between that of the dry adiabats 
and the vapor lines. See fig. 3-3 1. 

If the reader is interested only in an approximate equation for these 
adiabatic processes, he should go to section 3-34 at once. 

3-31. Exact equation of the pseudo-adiabatic process. To derive the 
exact equation for the pseudo-adiabats, let the saturated air be in the 
state (T,p,w s ) represented by A in fig. 3-31. After a small pseudo- 
adiabatic expansion, the air is in the state (r+ dT,p + dp,w 8 + dw 8 ) 
represented by B in the figure. Note that dT, dp, and dw 8 are all nega- 
tive. Let us consider a mass of 
l + w 8 tons of moist air, made up of 
one ton of dry air and w a tons of 
water vapor. In the pseudo-adia- 
batic process AB, the quantity dw 8 p + dp- 
of water vapor condenses and drops 
out as precipitation. The conden- 
sation releases the quantity of heat 

dll = -Ldw 8 , 



p, 




FIG. 3-31. 



(1) 

which is used to heat the moist air. 

From 2-22(3) the heat dh absorbed 

by the moist air per unit mass is related to the temperature change dT 

and pressure change dp as follows: 

(2) dh=c p dT-RT - 

P 

Here c p and R are the thermal constants for moist air. Since the mass 
of moist air is 1 + w 8 , we see that 

(l + w.)dh. 



(3) 

Combining (1), (2), and (3) we have 



(4) 



Ldw, i 



Section 3-31 70 

Equation (4) is an exact form of the differential equation for the pseudo- 
adiabatic process. 

It is convenient to express (4) in terms of the constants c p d and Rd 
for dry air. We substitute for R from 3-25(2) and for c p from 3-26(3). 
Multiplying through by (1 + w a ) and ignoring squares of w s , we get 

(5) -Ldw, - (1 + 1.90w 9 )cp4lT - (1 4- 1.61w 8 )RdT 

P 

Equation (5) can be shown to be equivalent to that derived by Fjeldstad. 

3*32. Exact equation of the reversible saturation-adiabatic process. 
As stated in 3*30, in the reversible saturation-adiabatic process the con- 
densed water is retained in the system in the form of cloud droplets. 
Let w be the total mass of water substance in a saturated parcel contain- 
ing unit mass of dry air. The system will then consist of 1 -f w 8 tons of 
moist air and w - w 8 tons of liquid water. Let the air be in the state 
(T,p,w 8 ) represented by point A in fig. 3-31. Let the expansion to the 
state (T+dT,p + dp,w 8 + dw 8 ) take place. As in 3-31 the quantity 
dw 8 tons of vapor will condense and release the quantity of heat 

(1) dlli - -Ldw.. 

A second source of heat is the cooling of the w - w 8 tons of water through 
dT degrees. This provides the quantity of heat 

(2) dH 2 = -c w (w - w 8 )dT, 

where c w is the specific heat of water. The total heat dHi 4- dH 2 
released by the process is absorbed by the moist air. For the 1 + w a 
tons of moist air this heat is equal to 

(3) (l + w 

Equating the total heat released in (1) and (2) to the heat absorbed in 
(3), we get 

-Ldw 9 - c w (w - w 9 )dT - (1 + w 8 ) c p dT - RT I 
Introducing Rd and c p d and transposing the term -c w (w - w e )dT give 



-Ldw 8 - [" 
L 



l + 1.90w 9 + (w- w.) c pd dT- 

P 



Putting in the value c w /c pd - 4185/1004, we get the final form 
(4) -Ldw 8 - [1 + 1.90nr. + 4.17(w - w 8 )]c pd dT- (1 + 



71 Section 3-34 

Equation (4) is an exact form of the differential equation for the reversi- 
ble saturation-adiabatic process. It will be noted from (4) that there is 
a different reversible saturation adiabat through (T,p) for each value w 
of the total water content. They differ only slightly from each other and 
from the pseudo-adiabat. 

3-33. Critique of the two equations. Equations 3-31 (5) and 3-32(4) 
cannot be integrated directly in their present form. The latter can be 
written in terms of the variables w 8 , /></, and T and is directly integrable 
in that form, a fact of questionable practical value, since pd is not a con- 
venient variable to use. The former equation seems non-integrable in 
terms of elementary functions, a consequence of its representing a non- 
adiabatic process. 

Either equation may be integrated numerically to any desired degree 
of accuracy by a series of small steps. To do this, a second relation 
between dw SJ AT, and dp is obtained by differentiating 3*20(4), which 
expresses the physical condition that the air remains saturated. See 
5-10(6). This second relation may be combined with the equation of 
either saturation process, and the two can be solved simultaneously 
for dT and dp in terms of dw 8 . The work is very laborious in practice. 

This numerical integration shows that the pseudo-adiabatic process 
cools slightly faster than the reversible saturation-adiabatic process. 
This is due to the loss of the heat content of the precipitated water. The 
difference is very slight and, in comparison with the effects of radiation 
and turbulent exchange of heat, may be neglected in practical problems. 

It is therefore immaterial for practical purposes whether the adia- 
batic process of saturated air is calculated from the reversible satura- 
tion-adiabatic or from the pseudo-adiabatic equation, provided the 
process is an expansion. The practical difference between the two 
appears when the process is reversed. When the condensed water 
remains in the air, the process is reversible, and the compression returns 
along its path of expansion. However, when a pseudo-adiabatic expan- 
sion is followed by compression, the compression nearly follows a dry 
adiabat. 

3*34. Simplified equation of the adiabatic process of saturated air. 
Let a parcel of 1 -f- w 8 tons of saturated moist air be in the state (p,T,w 8 ) 
represented by A in fig. 3-31. The parcel thus contains one ton of dry 
air and w 8 tons of water vapor. After a small adiabatic expansion the 
air is in the state (T+ dT,p -f dp,w 8 + dw 8 ) represented by B in that 
figure. Note that dT, dp, and dw 8 are all negative. 

Now the condensation of dw 8 tons of water vapor will release the 
quantity of heat Ldw 8 . Let us make the slightly incorrect assumption 



Section 3-34 72 

that this latent heat is used exclusively to heat the ton of dry air; i.e., 
we ignore the heating of the w a tons of water vapor. Then by 2-22(3) 
the heat dh absorbed by the dry air is related to the temperature change 
dT and pressure change dp as follows: 



dh = c pd dT - 

P 

Equating dh to Ldw a , we arrive at an important approximate equation : 
( 1 ) -Ldw B = c pd dT - R d T - '' 

It will be observed that the exact equations 3-31(5) and 3-32(4) both 
reduce to (1) when the small correction factors to c p a and Rd are neg- 
lected. The solutions of (1) are found to lie very close to the exact solu- 
tions of 3-31 (5) and 3-32(4). In view of the element of uncertainty in 
atmospheric problems, we are quite justified in using the equation (1) 
as an acceptable formula for the adiabatic process of saturated air. This 
will be done, and the process described by (1) will in the following be 
called the saturation-adiabatic process. The corresponding lines on the 
diagram will hereafter be called saturation adiabats. It will be shown in 
section 3-36 how these lines are constructed on the diagram. 
From 226(3), we may write (1) in the form 

Ldw s dO 

(2) - -*7* 

From (2) the change dO of potential temperature in fig. 3-31 can be 
expressed in terms of dw s . 

3 -35. Isobaric warming and cooling. For two later sections (3-36 
and 3'39) it is necessary to compute the temperature change resulting 
from isobaric evaporation from, or condensation into, a parcel of air. 
Let a parcel of moist air, saturated or not, be in the state (T,p,w). 
Suppose that some vapor is condensed from the parcel, or that some 
water is evaporated into the parcel. Let either process take place at 
constant pressure, the latent heat being supplied to or taken from the air. 
In the case of condensation the resulting change dw of mixing ratio is 
negative, and the air absorbs the latent heat by warming. In the case of 
evaporation dw is positive, and the air provides the latent heat by cool- 
ing. Let (dT) p be the resulting isobaric change in the temperature of the 
air. The air thus finishes in the state [T + (dT) p ,p,w 4- dw]. 

The expression giving (dT) p in terms of dw is easy to obtain. In the 
case of condensation, the latent heat made available to the moist air (per 



73 Section 3-36 

ton of dry air) is -Ldw. We shall, as in 3-34, assume that this heat is 
used exclusively to heat the ton of dry air. Since the process is isobaric, 
the heat dh absorbed by the air is from 2-22(3) 

dh = c p d(dT) p . 

Equating the heat Ldw released to the heat dh absorbed by the air, we 

get 

(1) -Ldw = c pd (dT) p . 

It will be seen that (1) is valid for either condensation or evaporation. 
It should be noted that formula (1) is mathematically equivalent to the 
special case dp = of formula 3-34(1). 

For easy computation, we solve (1) for (dT) p and substitute for L and 
c p d their values from 3-08(4) and 2-21 (5). The result is 

(2) (dT) p = -2.5(l(AfaO. 

By grouping 10 3 with dw we have expressed dw in parts per thousand, as 
on a diagram. Equation (2) can thus be expressed in words: 

At constant pressure, adiabatic condensation of one part per thousand of 
vapor will warm moist air two and one-half degrees. At constant pressure, 
adiabatic evaporation of the same amount of water into air will cool the 
air two and one-half degrees. 

3-36. Graphical construction of the saturation adiabats. The con- 
struction of the saturation adiabats can be carried out on a diagram very 
quickly by using 3-35(2). The method is equivalent to a numerical inte- 
gration of the equation 3-34(1). Suppose saturated air is in the state 
(T,p,w s ) represented by A in fig. 3-31. Let dw s be fixed at some con- 
venient small negative value, generally -1 X 10~ 3 or -2 x 10~ 3 . It is 
desired to find the point B (see fig. 3-31) where the saturation adiabat 
through A crosses the vapor line w 8 + dw 8 . For this construction we 
replace the saturation-adiabatic process AB by another adiabatic process 
A A f B consisting of two parts : (i) The latent heat released by the conden- 
sation of -~dw 8 tons of vapor is used to warm the air at constant pressure 
along the path AA f \ (ii) the now unsaturated air is brought back to 
saturation by a dry-adiabatic expansion A *B. Both (i) and (ii) are 
easily performed graphically. 

According to 3-35(2) the warming .4.4' will amount to 2.5 degrees for 
each part per thousand of vapor condensed. Thus A r is easily plotted. 
The point B is found at the intersection of the dry adiabat through A f 
with the vapor line w 8 -f dw 8 . When B has been obtained, the same pro- 
cedure can be repeated from that point, and so on. We thus obtain a 
series of points^, B, C, D, on the saturation adiabat. A smooth 



Section 3-36 



74 



curve can be put through these points for as long a distance as we choose 
to carry out the process. 

The approximation 3-35(2) is consistent for infinitesimal dw a with the 
approximate equation 3-34(1) defining the saturation-adiabatic process. 
The adiabatic process A A *B described above is therefore equivalent to 
the saturation-adiabatic process AB of 3 '34, as long as dw s is infinitesi- 
mal. For finite values of dw a , however, the method of the present sec- 
tion has a small error. The smaller the numerical values chosen for 
dw 8 , the more accurate the method is in practice. In this way it is 
limited only by one's ability to read the diagrams. 




FIG. 3-36. 

As an example we will compute the saturation adiabat through 
T 20C, p = 100.0 cb. This and other saturation adiabats are shown 
on the tephigram in fig. 3-36. A calculation yields w 8 = 14.9 X 10~~ 3 . 
The remaining steps have been done on a large tephigram and are shown 
in table 3-36. The student should follow the calculations on some 
legible diagram. 

There will be discrepancies on another diagram, of course, because of 
the human factor and the fact that few meteorological diagrams are 
made from strictly accurate plates. The point is that this method is 



75 Section 3-37 

convenient and reliable. More will be said in 3-38 about the asymptotic 
dry adiabat mentioned in table 3-36. It is labeled 6 = 6 e . 

TABLE 3-36 
POINT r(C) p(cb) 10 3 w, 10*dw a (dT) p 0(K) 

A 20.0 100.0 14.9 -0.9 2.25 293.0 

A* 22.2 100.0 295.2 

B 18.4 95.8 14.0 -2.0 5.0 295.2 

B' 23.4 95.8 300.4 

C 14.4 85.8 12.0 -2.0 5.0 300.4 

C' 19.4 85.8 305.7 

D 9.8 76.3 10.0 -2.0 5.0 305.7 

D' 14.8 76.3 311.2 

E 4.9 67.7 8.0 -2.0 5.0 311.2 

E' 9.9 67.7 316.8 

F -1.0 59.2 6.0 -2.0 5.0 316.8 

F f 4.0 59.2 ,. 322.8 

G -8.4 50.2 4.0 -2.0 5.0 322.8 

G' -3.4 50.2 329.2 

H -19.7 40.3 2.0 -1.0 2.5 329.2 

H' -17.2 40.3 332.6 

7 -29.2 34.0 1.0 -0.5 1.25 332.6 

/' -28.0 34.0 334.6 

/ -37.8 29.5 0.5 -0.5 1.25 334.6 

f -36.5 29.5 336.2 

(asymptotic dry adiabat: 6 = 336.2) 

3 -37. Nomenclature. A number of temperatures and potential 
temperatures of moist air are used by meteorological writers. There is 
fair agreement on the definitions of the variables to be introduced, but 
there is very little agreement on the names to be given them. It is 
therefore important to learn the variables in terms of definite operations 
on a diagram. By means of the operations themselves it is possible to 
distinguish what an author means by a certain complicated name. 

Concerning our own terminology the word wet bulb or subscript w 
always refers to temperatures attained by a parcel after it has been 
completely saturated. The word equivalent or subscript e always refers 
to temperatures attained by a parcel after it has been completely dried 
out. The word potential (or letter 6) always refers to a temperature 
attained after some kind of adiabatic compression to 100 cb. 

Any temperature with the prefix isobaric or subscript i is that attained 
by a parcel after being saturated or dried out at constant pressure. 
These are in contrast to temperatures labeled by a prefix adiabatic or 
subscript a. The latter are temperatures attained after a parcel has been 
saturated or dried out along dry and saturation adiabats. 



Section 3-38 76 

3-38. Definitions of O w , O e , T aw , T aeJ T d . The method of labeling dry 
adiabats can be applied to the saturation adiabats. Every saturation 
adiabat intersects the isobar p = 100 cb. The value of the temperature 
at this intersection is called the wet bulb potential temperature, and is 
denoted by 6 W . See fig. 3-38. The value of 6 W uniquely labels the 
saturation adiabats. On many diagrams the saturation adiabats are 
drawn at intervals for O w of 2C. 

A parcel of moist air is said to have the wet bulb potential temperature 6 W 
of the saturation adiabat through the characteristic point (see 3-28) of the 
parcel. 




FIG. 3-38. 

Another way of labeling the saturation adiabats is obtained from the 
fact that as w 8 approaches 0, the saturation adiabat approaches asympto- 
tically a certain dry adiabat. (This is not proved here but is plausible 
from a diagram.) The potential temperature of this asymptotic dry 
adiabat is called the equivalent potential temperature and is denoted by B e . 
This value of 6 e uniquely labels the saturation adiabat. On some dia- 
grams the value of O e is given at the low-pressure end of each saturation 
adiabat. On other diagrams e must be computed according to the 
method described in 341. 

A parcel of moist air is said to have the equivalent potential temperature e 
of the saturation adiabat through the characteristic point of the parcel. 



77 Section 3-39 

The two potential temperatures O w and Q e are both used in the litera- 
ture and both must be familiar to the meteorologist, even though either 
one of the two would be quite sufficient in theory. 

Consider any parcel of moist air with pressure p, wet bulb potential 
temperature O w , and equivalent potential temperature 6 e . The tempera- 
ture at the point where the saturation adiabat 6 W crosses the initial 
isobar p is called the adiabatic wet bulb temperature and is denoted by 
T aw - See fig. 3-38. The temperature at the point where the dry adiabat 
6 = e crosses the initial isobar is called the adiabatic equivalent tempera- 
ture and is denoted by T ae . 

The dew point temperature Td of a parcel of moist air is the temperature 
at which the air would become saturated if it were cooled isobarically 
without change of mixing ratio. This temperature is that of the inter- 
section of the initial isobar with the vapor line w 8 = w. See fig. 3-38. 
The cooling process defining Td is not an adiabatic process. 

Each of the temperatures just defined has found theoretical and practi- 
cal applications to meteorology, and should be known. Each can be 
defined by an ideal physical process on the initial parcel, and the student 
should have no trouble in stating the processes needed. For example, 
T a e is attained by a parcel from the successive application of three 
processes: (i) expanding the parcel by a dry-adiabatic process until it 
becomes saturated at the characteristic point; (ii) completely drying 
the parcel out by a saturation-adiabatic expansion to very low pressure; 
(iii) compressing the resulting dry air dry adiabatically to the initial 
pressure. 

3 '39. Definitions of T ie and T iw . Consider the fictitious process 
where a parcel of moist air is dried out by an isobaric process which is 
also adiabatic. That is, we suppose that all the water vapor is condensed 
out of the parcel, the latent heat released being used to heat the air 
itself. The resulting temperature is called the isobaric equivalent tem- 
perature and is denoted by T ie . It can be shown that for moist air we 
always have 

(1) ' T ie < T ae . 

Both the processes which define T ae and Ti e are adiabatic, and the 
amount of latent heat released is the same in both of them. The final 
temperatures are, however, different, since the two processes follow 
different paths in the diagram and therefore perform different amounts of 
work. 

In contrast to the temperatures defined in 3-38, which are usually 
obtained graphically, Ti e is obtained by a simple calculation. From 
3-35(2) each part per thousand of vapor condensed will heat the air 



Section 3-39 78 

2.5C. Hence with good accuracy, 
(2) r ie 



where w is the mixing ratio of the air. 

The corresponding wet bulb temperature is obtained from the initial 
parcel by evaporating water vapor into the parcel isobarically and 
obtaining the necessary latent heat by cooling the parcel. This process 
may be assumed to take place when precipitation falls through a layer 
of unsaturated air. Thus the mixing ratio rises from its initial value w, 
while the temperature falls from its initial value T. The temperature 
where the air becomes saturated is called the isobaric wet bulb temperature 
and is denoted by T^ w . The corresponding saturation mixing ratio is 
denoted by Wi W . It can be shown that for unsaturated air we always 
have 

1 aw < * iwi 

for the same reason as in the case of the equivalent temperatures. The 
difference T^ T a \o rarely exceeds a few tenths of a degree. For this 
reason many authors lump the two into a single " wet bulb tempera- 
ture." 

TW can be very accurately obtained by a combination of reading a 
diagram and numerical interpolation, as follows. Integrating 3-35(2) 
between the initial state and the saturated state, we get 



T iw -T -- 2.5 x 10 3 (w t - w - w). 
This may be written in the form 
(3) T iw + 2.5(10 3 w iw ) - T+ 2.5(10 3 ze/) - T ie . 



Since both Ti W and Wi W are unknown, (3) cannot be solved directly. 

For any temperature T on the initial isobar, the corresponding satura- 
tion mixing ratio w a can be read from a diagram. Then the variable 
quantity 



can easily be computed. Now the solution Ti w of (3) is the value of T 
such that aT = 7\ e . This is most easily found by choosing a T likely to 
be a little too low, for example T aw , and another T likely to be a little 
too high. On computing a^p for both of these guesses, a direct interpola- 
tion will yield the T for which aT = Tie- This T is T{ w . All tempera- 
tures may be expressed in centigrade in this computation. 

A more important problem in practice is to obtain w, being given Tand 
Ti W . This is much easier, since from a diagram we can get Wi W at once. 



79 Section 3-41 

Then (3) can be solved for w, yielding the so-called psychrometric formula 
(4) 



The importance of this formula will appear in the next section. 

It can be shown that the temperatures defined so far stand in the 
following relation for moist unsaturated air: 

T 8 <T d < Taw < T iw <T< Tie < Tae- 

This can be used as a check on the computations. 

3 '40. Surface observations. The humidity is obtained in surface 
observations by measuring the " wet bulb temperature " with a 
psychrometer. The evaporation from the damp cloth lowers the 
temperature by a very complex process, which probably cannot be ade- 
quately described. Though it is an isobaric process, it certainly is not 
adiabatic, so that the temperatures of 3-39 do not strictly apply. How- 
ever, it is found that with very little error we can assume the wet bulb tempera- 
ture to be equal to Ti w . 

With this assumption, we can regard the raw data of a surface obser- 
vation as being the values of T, Ti w , and p. From the psychrometric 
formula 3-39(4) we can get w. Thus we have the state of the air in 
terms of T, p, and w. From this point on all other desired quantities can 
be obtained as outlined previously. 

In particular: r can be obtained from 3-21(2) ; T*, from the method of 
3-38; and e, from the method of 3-23. These are the moisture variables 
usually reported in a surface observation and obtained from psychro- 
metric tables. The present discussion has shown how a surface observa- 
tion report can be prepared independently of such tables by methods 
which are rapid and practical, requiring only a diagram. 

3-41. Example. The following example will show how the formulas 
and processes described in this chapter are used to evaluate the different 
variables. The graphical operations are done on a tephigram like that 
of fig. 3-36. 

Given air in the initial state T - 13.5C, p - 90.0 cb, w - 3.0 X 1(T 3 . 
The following quantities are obtained immediately by calculation: 
T*,R, c p , *, a: 

By 3-29(1), r* - 13.5 + }3 - 14.0C - 287.0K. 
By 3-29(2), R - 287 + }3 - 287.5 kj t" 1 degf 1 . 
By 3-29(3), c p - 1004 + A3 = 1006.7 kj t" 1 deg~ l . 
By 3-29(4), K - 0.286 - 3(0.0001) = 0.286. 
By 3-25(5), a - (287)(287)/(90) - 915 m 3 t" 1 . 



Section 341 80 

The following quantities are usually obtained from a diagram : w a , 8, 

Pai * 81 v wt 1 oiy , C , 1 aei 1 di &S- 

Plotting the image point (13.5C, 90.0 cb) on a diagram, we read 
W 8 =11.0xl0~ 3 and we interpolate 6 = 295. 5K between the dry 
adiabats. Following this same interpolated adiabat to the point where 
it intersects the vapor line w 8 = 3.0 x 10~ 3 , we read the pressure p s = 68.3 
cb and temperature T 8 = -8.2C. This point is the characteristic 
point. The saturation adiabat through this point intersects the 100-cb 
isobar at the temperature 6 W = 10.0C = 283.0K. 

Going back up the same saturation adiabat, we read the temperature 
T aw where it crosses 90.0 cb, getting T aw = S.OC. We continue on the 
saturation adiabat to the point ( -24.9C, 51.2 cb) where it crosses the 
vapor line w a = 1 X 10~~ 3 . We get d e by following the construction of 
section 3-36. Since W 3 w s = 1, we add 2.5C isobarically to the last 
point mentioned and get ( -22.4C, 51.2 cb). We read the potential 
temperature O e = 304K. (This is the only safe method on most dia- 
grams, as it is impossible to estimate which dry adiabat the saturation 
adiabat is approaching.) Following the dry adiabat 6 = O e to 90.0 cb, 
we read T M - 21.8C. 

We now read the dew point temperature Td = -4.7C at the intersec- 
tion of the vapor line w s 3.0 X 10~ 3 with the initial isobar p = 90.0 cb. 

To get e 8 , we follow the isotherm T = 13.5C to its intersection with 
the pressure 622 mb (see section 3-23). Here 10 3 w = 16, which when 
added to 622 gives 638. At (13.5C, 638 mb) we read 10X' - 15.5, 
so that e a = 15.5 mb = 1.55 cb. 

At this stage, we can obtain r, e, Ti e , and Ti w by calculation. By 
3-21(2), r w/w a = 3.0/11.0 = 27%. Then by 3-19(2), e = re, = 
(0.27)(1.55) - 0.418 cb. From 3-39(2), T ie - 13.5 + 2.5(3) - 21.0C. 
To get T iw we follow the method of section 3-39. We use T aw = 5.0C 
as the first guess. From the diagram at (5.0C, 90.0 cb), w 8 - 6.1 x 
10~ 3 . Thus 

a 5 = 5.0 + (2.5) (6.1) - 20.25. 

Similarly, trying 6.0C as a second guess, with the corresponding value 
w 9 - 6.5 X 10"" 3 , we have 

a 6 = 6.0 + (2.5) (6.5)- 22.25. 

Interpolating between 5 and 6 to get ar - T ie - 21.0, we get 

21.0 -20.25^ 
22.25- 20.25 ~ 5-4U 

Several of the graphical operations can be checked by use of formulas, 
if desired, but unless exceptional accuracy is desired this is unnecessary. 



CHAPTER FOUR 
HYDROSTATIC EQUILIBRIUM 

4-01. The hydrostatic problem. The preceding chapters have 
treated the physical behavior of individual air elements. We shall now 
proceed to discuss the distribution of air elements in space. In the 
present chapter the case will be considered where the atmosphere is in 
equilibrium. Although this case never holds in practice, there are 
several reasons why it is important to examine the atmosphere in this 
state. First, it serves as the natural introduction to the more compli- 
cated general problem of the atmosphere in motion. Second, the analy- 
sis of a resting atmosphere will provide useful insight into the general 
laws for the distribution of mass and pressure. Third, it will be shown 
later (see 7-14) that an important practical problem can be solved with 
sufficient accuracy by assuming the atmosphere at rest. 

This so-called hydrostatic problem consists in determining the dis- 
tribution in space of the physical properties of the air elements, i.e., their 
pressure, temperature, specific volume, and density. Our knowledge of 
this distribution is in practice obtained by a series of meteorological 
observations, which are taken simultaneously at the surface of the earth 
and, to a less extent, from the free atmosphere. The surface observa- 
tions give values of the atmospheric variables at fixed points. The 
upper air observations on the other hand are made by sounding instru- 
ments which give simultaneous values of the atmospheric variables, 
pressure, temperature, and humidity. The soundings give no direct 
information about the position in space where the various values occur. 
Thus the problem to be solved in order to determine the distribution in 
space of mass and pressure is to refer the upper air observations to geo- 
metric points, or more specifically to determine the heights which corre- 
spond to the various pressure values along an aerological ascent. The 
solution of this problem is the practical aim of this chapter. 

4 '02. The fields of the physical variables. Each geometrical point 
within the region of the atmosphere is occupied by an infinitesimal ele- 
ment of air in a certain physical state, characterized by certain definite 
values of the physical variables p, a, T. A region where every point has 
a definite value of a physical property assigned to it is known as a field 
of that property. 

81 



Section 4-02 82 

Let us consider any one of the physical variables, for instance, the 
pressure. Apart from certain singular points the pressure will from 
every point in space increase in certain directions and decrease in others. 
Thus, through every point there is a surface on which the pressure does 
not change, which separates the region of increase from the region of 
decrease. This surface does not terminate anywhere in the interior of 
the atmosphere; it either is closed or continues until it intersects the 
surface of the earth. This surface is called an isobaric surface. It 
divides space into a region of higher pressures on one side and a region of 
lower pressures on the other side. Consider now two isobaric surfaces 
characterized by the pressures p and p + S/>. These surfaces do not 
intersect, for the pressure has only one value at a point. The normal 
distance between the two surfaces is everywhere small if the pressure 
interval dp is sufficiently small. The two surfaces bound a thin layer 
which is known as an infinitesimal isobaric layer. The whole atmosphere 
may accordingly be divided into a large number of thin isobaric layers. 
In practice it is unnecessary to draw many surfaces, since a small number 
gives a sufficiently clear picture of the pressure field. Generally the sur- 
faces p = 0, 1, 2, cb are selected. The layers defined by these sur- 
faces are called the isobaric unit layers. This method of representation is 
essentially the one used in practical meteorology. The isobars which 
are drawn on the synoptic weather map are the intersections between the 
isobaric surfaces and mean sea level. If similar isobars are drawn in a 
number of level surfaces sufficiently close together, one obtains in prac- 
tice the three-dimensional picture. We shall in the following discussion 
only occasionally make use of the two-dimensional isobars in the level 
surfaces, and generally think of the pressure field in terms of its isobaric 
unit layers. 

A similar geometrical representation of the temperature field is given 
by the isothermal surfaces and the isothermal unit layers. The mass 
field may be represented either by the specific volume or by the density. 
According to 2-03(1), a surface a = const is always a surface p = const, 
so these two variables define the same family of surfaces. However, the 
selection of unit surfaces and unit layers is different, depending upon 
which of the two is used for the description of the mass field. When the 
specific volume is used, the mass field is geometrically represented by the 
isosteric surfaces and unit layers. The surfaces and unit layers of den- 
sity are called isopycnic. 

Any physical quantity whose field can be described by one single 
numerical value in each point in space may be given a similar geometrical 
representation. There is no essential difference between such fields, and 
accordingly they may be treated mathematically as quantities of the 



83 Section 4-04 

same kind. It is then unnecessary to derive the mathematical rules for 
such quantities more than once. Physical quantities of this kind are 
known in mathematical physics as scalars, and their fields as scalar fields. 
We shall presently meet other physical quantities which for their descrip- 
tion require more than one number in each point. Before these quanti- 
ties are discussed we shall introduce analytical expressions for the 
scalars. 

403. The coordinate system. The fundamental tool which is used 
to describe geometrical quantities in analytical terms is the coordinate 
system. The ideal system for atmospheric problems would of course be 
one with spherical coordinates. For most purposes, however, the Car- 
tesian or rectangular system is satisfactory, and is preferable because of 
its great simplicity. The Cartesian system generally adopted in mathe- 
matical physics is the so-called right-handed system, which by definition 
is such that a rotation in the 
xy plane from the positive x . 

axis to the positive y axis will 
drive a right-handed screw in 
the direction of the positive z 
axis. See fig. 4-03. It then 
follows that the rotations y to 



z and z to x will also give the 




(north) 



right-handed screw displace- 

ments along the positive x 

axis and y axis respectively. x (east) 

We shall later apply this FlG> 4 . 3. The standard system of coordinates. 

right-handed screw rule when- 

ever the rotation in a plane is associated with a positive direction along 

the axis of rotation. 

In meteorology this system of coordinates is generally given a fixed 
position and orientation with reference to the earth. The origin is 
placed at some special point, fixed in the earth's surface. The xy plane 
is chosen as the plane tangent to the surface of the earth at the origin, 
with the x axis to the east and the y axis to the north. This choice 
automatically fixes the direction of the z axis as pointing vertically up- 
ward. This system will be referred to as the standard system. 

4*04. Analytical expression for scalar quantities. It was shown in 
section 4*02 that any scalar quantity may be represented geometrically 
by its surfaces of constant value of the scalar, or more specifically by its 
equiscalar unit layers. Let the scalar be denoted by e, and consider first 
one of its surfaces defined by the value ei. The equation for this surface 



Section 4-04 84 

in a rectangular system of coordinates will be of the form ei = 
where &(x,y,z) is a certain definite mathematical function of the variables 
x, y, z. If this function is put equal to another constant value e 2 close to 
61, we obtain the equation for another surface: 2= *(x,y,z), which 
generally is close to the surface ei , and so on. Thus the whole scalar field 
may be described analytically by the expression 

(1) e=e(w). 

By successively giving e the values 0, 1, 2, this expression gives the 
equations for the equiscalar unit surfaces. 

The expression (1) cannot, of course, be specified further, since it repre- 
sents all possible functions. Under real conditions in the atmosphere the 
distribution and shape of the equiscalar surfaces of the physical variables 
are in general so complicated that it is impossible to express the corre- 
sponding functions (1) in terms of explicit mathematical expressions. 
Nevertheless we know that the functions exist, and that they are con- 
tinuous and single valued in space, except along frontal surfaces. This 
information is sufficient for the theoretical analysis of their dynamical 
behavior. 

The fields of the three scalar quantities with which we are primarily 
concerned are expressed mathematically as follows: 

(2) p - />(*,?,*); - (*,?.*); r- r(* f y f ). 

These expressions, or the equivalent geometrical representation by unit 
layers, define the fields of these quantities at a certain fixed time. In 
the case of equilibrium this representation is complete. If the atmos- 
phere is in motion, however, the fixed points (x,y,z) will continually be 
occupied by new particles, bringing their physical properties along with 
them. For each new instant we would have a new distribution in space 
of the unit layers and a new set of functions (2). This is analytically 
expressed by introducing the time t as an additional variable, which 
gives : 

(3) p . p(x,y,z,t) ; a = a(x,y,zj) ; T T(x,y,z,t). 

When t here is given a constant value /o. we are back to the expressions 

(2) for a given instant. When / changes, the expressions (3) describe 
how the surfaces change their position with time. 

4-05. Vectors. The physical condition for the equilibrium of a 
system is that the forces acting upon it are in balance, or that their 
resultant is zero. Thus the study of equilibrium brings in a new physi- 
cal concept, force. Force belongs to a class of physical quantities differ- 



85 Section 4-05 

ent from the scalars. Whereas a scalar is characterized by magnitude 
only, and can be represented by one number when the units are chosen, 
force has direction as well as magnitude. Other physical quantities 
belonging to this class are displacement, velocity, acceleration, etc. 
These directed quantities are known as vectors. 

Any vector may be represented geometrically by an arrow, pointing 
in the assigned direction, whose length is equal to the magnitude of the 
vector. Since all vectors thus have the same geometrical representation, 
the mathematical rules which apply to one of them apply to them all. 
These rules may be developed without specification of the physical nature 
of the vectors. 

The vectors will in the following be denoted by bold-face letters, and 
their magnitudes by the corresponding letters in ordinary type. Thus if 
a vector is denoted by a, we have by definition |a| = a. We shall often 
deal with vectors of unit length. Thus if 1 is a unit vector we have by 
definition |l| = 1. 

Consider an arbitrary vector a and an arbitrary straight line / with an 
assigned positive direction determined by a unit vector 1 (fig. 4'05). It 
is always possible to pass planes normal to / through the two endpoints of 
the vector a. The projection of the vector on the line, defined by this 
operation, is called its component 
along the direction / and will be 
denoted by ai. The component 
is a pure number, and hence a 
scalar. It is positive when the 
projection extends in the direc- 
tion of 1, and negative when it \ 
extends in the opposite direc- 
tion. When the operation illus- 
trated in fig. 4'05 is applied to the three axes of the coordinate system we 
obtain the rectangular components a x , a y , a z of the vector a. 

If e is a scalar and a is a vector, the expression ea is used to represent 
a vector e times as long as a, having the same direction if e is positive, the 
opposite direction if e is negative. In particular, if e = -1, then ea = -a 
is a vector equal in magnitude to a, but having the opposite direction. 
If A is a unit vector in the direction of the vector a, then a = aA. It is 
important to note that the component a\ is not a vector, but that 
a /I is a vector. 

Before we take up the discussion of the state of equilibrium, it is 
necessary to derive a few simple mathematical rules for vectors which 
will be needed in this discussion. The immediate need will be the 
expression for the resultant of the acting forces. 




Section 4-06 



86 



406. Vector sum. The resultant of two forces acting on the same 
particle is by physical definition a third force whose effect on the particle 
is identical to the joint effect of the two forces. Experiments show that 
when the two forces are represented by arrows (fig. 4*06a), their result- 
ant is geometrically represented by the diagonal in the parallelogram 




FIG. 4-06a. 



defined by the two arrows. This law of the parallelogram is also valid 
for displacements, velocities, accelerations, etc., or more generally for all 
vectors. 

The parallelogram is conveniently replaced by the triangle (fig. 4-066) 
defined by the two vectors. Their resultant is the third side of this tri- 
angle. We shall introduce for the resultant c of the vectors a and b the 
notation 

(1) c 



and call c the vector sum of the two vectors. Similarly the sum of three 
or more vectors is obtained by constructing a polygon of the vectors, 
and drawing a vector from the initial point of the first to the terminal 
point of the last. By inspecting this polygon it is easily verified that 
the commutative and associative laws are valid for vector addition. 

If the polygon representing the sum of two or more vectors is closed, 
the vector sum is the zero-vector, denoted by 0. Thus the vector sum of 
a and -aisO: 

- a - a. 

The zero-vector, being a vector of zero magnitude, should be distin- 
guished from the scalar zero. 

An important example of the vector sum is the representation of a 
vector a in a rectangular system of coordinates. The three unit vectors 
along the positive axes of x, y, z will be denoted by i, j, k. If the opera- 
tion illustrated in fig. 4-05 is applied to the three axes of the system, we 
obtain the three vectors a x i, a v j, a 2 k. According to the above rule their 
sum is equal to the vector a; hence 



(2) 



a = a x i + a v j + o z k. 



87 



Section 4-07 



A similar expression may be derived for the vector b. When these 
expressions are introduced in (1), we obtain 



(3) 



c - a + b - (a x + b x )i + (a v + 6 y )j + (a, + b. )k. 



The rectangular components of a + b are obtained by adding corre- 
sponding components of a and b. This rule also follows directly from 
fig. 4-066, by projecting the triangle upon an arbitrary straight line / 
by planes normal to /. 

4*07. Scalar product. Another important vector operation is exem- 
plified by the element of work 2*14(1), where the two vectors in question 
are the force and the infinitesimal displacement. Let a and b be any 





a 

FIG. 4-07o. 



FIG. 4-076. 



two vectors, and let 6 be the angle between them (fig. 4-07a) . The scalar 
quantity ab cos B is defined as the scalar product or dot product of the two 
vectors, and is indicated by placing a dot between the factors. Hence by 
definition, 

(1) ab ab cos - ab a . 

In the third expression we have used the notation b a for the component 
of b along a. Thus the scalar product of two vectors is a scalar. Obviously 

a-b - b*a; 

i.e., scalar multiplication obeys the commutative law. 

Let b a and c a be the components of b and c along a. From fig. 4-076 
it is clear that 6 a -f c a is the component of b + c along a. From the 
equation 

(2) 



it follows by using the rule (1) on each term that 
(3) a*(b+ c) - a-b-f ac. 

This proves that the distributive law is valid for scalar multiplication. 

If two vectors are parallel and have the same sense, cos 6 1, and their 

scalar product is equal to the regular product of their lengths. If the 



Section 4-07 88 

two vectors are perpendicular, cos 6 = and their scalar product is zero. 
Conversely, if a scalar product is zero, one of the vectors is zero or else the 
two are perpendicular. 

Applying the definition (1) to the unit vectors i, j, k, we obtain 

( A\ H = i'i = k-k = 1 ; 

w i-j - j-k = k-i = 0. 

To obtain another expression for a*b, we first write a and b in compo- 
nent form, as in 4-06(2). Then, according to (3), we take the scalar 
product of each term of a with each term of b. Finally, applying (4), 
we see that 

(5) ab = a x b x -f a y b y + a z b z = ab cos 0. 

All three equivalent expressions in (5) for the scalar product are useful, 
according to the nature of the theoretical problem. We shall generally 
use the compact expression to the left, and only make use of the expanded 
forms in practical applications. 

4*08. Mechanical equilibrium. A system is in mechanical equilib- 
rium when the resultant of the forces acting upon it is equal to zero. 
Applied to the atmosphere, this principle can be formulated as follows : 
The atmosphere is in equilibrium when for any arbitrary part of the atmos- 
phere (not necessarily infinitesimal) the resultant of all the acting forces is 
zero. There are two sets of forces to be considered in the atmosphere: 
the external forces acting upon the air elements from without, and the 
internal forces arising from the interaction between the air particles 
themselves. The only external force in the atmosphere is the force of 
gravity. In the case of equilibrium the only internal force is the pressure 
force. In order to formulate the equilibrium condition mathematically 
it is necessary to know the analytical expressions for these two forces. 

4*09. The force of gravity. By the force of gravity we mean the 
force imparted by the earth to a unit mass which is at rest relative to the 
earth, i.e., which rotates with the earth. Since it is a force per unit 
mass, it has the dimensions [LT~ 2 ] of acceleration and indeed is some- 
times called the acceleration of gravity. The force of gravity is a vector 
denoted by g, whose magnitude is therefore g. 

If the earth had no rotation, the force of gravity would be identical to 
the force g a of pure gravitational attraction, directed toward the center 
of the earth. The earth would be a perfect sphere whose gravity would 
have the same magnitude everywhere on its surface. According to 
Newton's law of universal gravitation, the magnitude g a of the force of 
gravitation on a unit mass situated at the distance r from the earth's 



89 



Section 4-09 



center is K/r 2 . Here K is a constant fixed by the mass of the earth. 
See 6-08. 

Since the earth is actually rotating, the force observed as gravity is 
the resultant of g a and the centrifugal force arising from this rotation. 
(This will be discussed in more detail in section 6-10, in connection with 
motion relative to the earth. ) The centrifugal force is directed outward, 




FIG. 4-09. The force of gravity. 

normal to the axis of the earth. Its magnitude is numerically equal to 
Q?R, where ft is the angular speed of the earth and R is the distance from 
the axis. Let R be the vector from the axis of the earth, extending 
normal to the axis and ending on the point in question. Then the cen- 
trifugal force is 2 2 R. Fig. 4-09 is a schematic diagram of these forces 
in which the eccentricity of the earth has been exaggerated. 

The force of gravity g is the vector sum of g a and fi 2 R. It is impor- 
tant to note that the motion of the earth enters into the definition of 
gravity. Thus the equilibrium we are considering is no real equilibrium, 
but rather a state where the earth and the atmosphere rotate together as 
a rigid system. We shall call this state relative equilibrium. It will be 
explained in section 640 why this state can be investigated as though it 
were a real equilibrium. 

At present we wish to derive an expression for the dependence of g 
upon the latitude <p and height z above the surface of the earth. The 
angle between g and g is very small. Hence the magnitude of g can be 
closely approximated by subtracting from g a the length of the projection 
AB of fi 2 R on g a . But the length of AB is equal to ti?R cos <f>. See 
fig. 4-09. Since R = r cos <p, we can write 

(1) length of AB - Q 2 r cos 2 <p. 



Section 4-09 ^90 

As above, we have for the magnitude of g : 

(2) fo-pr 

Subtracting (1) from (2), we get an approximate expression for the 
magnitude g^ of the force of gravity at latitude ^: 

j/r 

(3) g v = ^ - fl 2 ' cos 2 ,. 

Letting 45 be the value of g at 45 latitude, we have, from (3), 

(4) g45=*-*fl 2 r. 

Eliminating K/r 2 between (3) and (4), we find 

(5) g v - g 45 - % 2 r cos 2<f>. 

Now r is approximately constant on the earth's surface. Hence the 
expression 12 2 r is nearly constant, and (5) suggests that a formula for g v 
should take the form 

(6) g v - 45(1 - a>\ cos 2?) (#i = const). 

To get an expression for the variation of gravity with elevation 0, 
we fix our attention on the poles, where the centrifugal force vanishes. 
At the height z above the pole, the distance from the center of the earth 
is r + 2, whence from (3) the gravity g z becomes 



At the ground we have the value g of gravity 

(9) go = ^' 

Dividing (8) by (9), we get 



Hence to a good approximation 



*o 

Since r is approximately constant, formula (10) suggests that a formula 
for gt at any height should take the form 

(11) gz - go(l - 02*0 tea - const). 



91 Section 4- 10 

Putting (6) and (11) together, we have a suggested type of formula for 
the value g^ tZ of gravity at latitude <p and elevation z: 

(12) g ViZ - g45,o(l - CL\ cos 2<p)(l - a 2 z). 

The actual best values of ai and a 2 in (12) can be found only by statisti- 
cal methods, based on measurements of gravity. These measurements 
are obtained from pendulum experiments all over the earth, and they 
yield the formula 

(13) g^z - 9.80617(1 - 0.00259 cos 2?)(1 - 3.14 x KT 7 z), 

where z is the elevation in meters. The value of a 2 in (13) can 
be obtained from (10), but the value of a\ is some 50% larger than that 
computable from (5). 

Denoting the sea level value of gravity at the pole by gp, and the value 
at the equator by g#, we get approximately in m s~ 2 : 

(14) flr-9.78; 45= 9.81; g P = 9.83. 

It is sufficient to remember these values for our later discussions. 

Now that the physical nature of the force of gravity has been estab- 
lished, our next task will be to derive a suitable analytical expression for 
it. Equation (13) cannot be used for this purpose, since it gives no 
information about its direction. The first step toward the analytical 
expression is to find a geometrical representation of the force of gravity. 
A very simple and convenient representation is obtained by means of the 
geopotential. 

4-10. The field of geopotential. Consider an isolated particle of unit 
mass moving through the field of gravity along an arbitrary path. Let 
dr be an infinitesimal vector displacement of length ds along this path, 
and let be the angle between the force of gravity and dr. The work dw 
of gravity on the particle during this displacement is, according to 
2-14(1) and 4-07(1), 

(1) dw = gds cos - gvfr. 

The work is positive if the angle is acute, and negative for an obtuse 
angle. This work can, according to the principle of the conservation of 
mechanical energy (section 2-18), be provided only through some 
expenditure of energy. Since the particle is isolated and undergoes no 
physical changes during the motion, the energy must be stored in the 
particle as a consequence of its position in the gravitational field. This 
form of energy which is released whenever a particle moves in the field 
of gravity is called potential energy. Denoting the change in the poten- 



Section 4- 10 92 

tial energy of unit mass during its displacement dr by d<t>, the conserva- 
tion of energy gives dw + d<t> = 0, which introduced in (1) gives 

(2) d<t> = -g'dr. 

We see that d<t> is an exact differential (section 2-27) by letting the particle 
return to its initial position after having moved in an arbitrary closed 
path. The final state is then in every respect identical to the initial, 
and the conservation of mechanical energy requires that the total 
amount of energy gained during this motion is zero. Thus is a function 
only of the position in space, i.e., a function of the coordinates x,y,z. 
Hence 

(3) * -*(*,?,*). 

We shall now introduce the letter 8 to denote differentials which repre- 
sent the difference in value of some physical variable between two points 
in space, the difference being computed at a fixed time. Differentials 
denoted with 6 are called " geometrical differentials." We shall reserve 
the letter d for differentials representing the change in value of some 
physical property of a particle during a process on that particle. Such a 
process will consume a certain amount of time dt, in contrast to 
the instantaneous nature of the geometrical differentials. Differentials 
denoted with d are called " process differentials." As an example of 
the distinction made here, the thickness of a falling leaf would be 62, 
but the distance the leaf dropped during a certain time would be dz. 

An example of the geometrical differential is the variation in between 
two points in space separated by a small distance. The distance is 
denoted by 5r, and the corresponding variation in is 50. We have from 
the above argument that 

(4) 60 = -g'6r. 

Equation (4) refers to as a function of position in space; equation (2) 
refers to the change of potential energy of a moving particle. Since 
is a function only of position in space, 50 is according to section 2-27 an 
exact differential of the three variables x,y,z. 

We notice that when the angle between g and 5r is acute 50 is nega- 
tive and the potential energy decreases in the direction of 6r; when is 
obtuse 50 is positive and the potential energy increases in this direction. 
For displacements normal to the force of gravity (6 90), 60 equals 0, so 
any such displacement is contained in a surface = const. These sur- 
faces, which are defined analytically by giving constant values to in 
(3), define the horizontal in each locality and are called level surfaces. 
To examine the variation in from level to level we consider a displace- 
ment in the direction opposite to gravity and hence normal to the level. 



93 Section 4- 10 

Denoting this displacement by Sz we have from (4), since cos 9 = 1 : 

(5) ' 50 = gdz. 

Thus the variation in < is equal to g per unit displacement normal to the 
level. Therefore if the potential energy is known at one level, its value 
can be computed for any other level by integration of (5). Let this 
known value at the reference level 3 be <fo- Its value at any other level 
z\ is then <i, where 



-o- J 



(6) 0i - 4> - / g8z. 



^ We are never concerned with absolute values of energy, but only with 
its variations. Therefore we may choose our reference level as we please 
and give this level an arbitrary value of potential energy. The simplest 
choice is the surface of mean sea level as reference level. We shall assign 
to this level the potential energy value zero. Once this choice has been 
made, the potential energy throughout space is prescribed and is given 
at the elevation z by the expression 



2 



(7) <t> = J gSz. 



The integration is taken along a vertical from sea level to the point in 
question, i.e., along a path normal to the level surfaces. The integral is 
easily evaluated when the expression 4-09(13) for g is introduced. To 
obtain the complete three-dimensional distribution of the integral 
must be computed for each latitude <p. The function in (3) may accord- 
ingly be considered a known function. Since this function is physically 
derived from the concept of potential energy in the field of gravity, it is 
known as the potential of gravity or the geopotential. We shall usually 
write it in the general form (3) and in our imagination represent this 
function by its surfaces for unit values of 4>. 

These surfaces not only give a clear geometrical picture of the func- 
tion 0. They also can be used to define the force of gravity. Denoting 
the thickness of the unit layer by h<j>, we have from (5), when the varia- 
tion of g through the layer is neglected, 



5z h+ 
From this expression and the previous discussion we may formulate the 

rx'kll/viirtfinr oirviril/a fiilA* 



following simple rule: 



Section 4-10 94 

The force of gravity is normal to the surfaces <t> = const, is directed toward 
decreasing values of <, and its magnitude is equal to the reciprocal of the 
thickness of the unit layer of </>. The gcopotential unit layers provide a 
convenient geometrical representation of the force of gravity. The 
corresponding analytical relation will give us the mathematical expres- 
sion for this force. Before proceeding to derive this expression it will be 
useful to examine the distribution of the geopotential unit layers quanti- 
tatively. 

4-11. Dynamic height. Geopotential was defined physically as the 
potential energy per unit mass. It is therefore measured in specific 
energy units (see table 1'05), and has the dimensions [L 2 T~~ 2 ]. In the 
mts system this unit is 1 kilojoule per ton. We can therefore give 
4-10(8) the following quantitative interpretation: When the level sur- 
faces are drawn at intervals of 1 kilojoule per ton, the thickness of the 
layers is h^ 1/g meter, or approximately 1 decimeter. This means 
that if 1 ton of mass is lifted this vertical distance its potential energy 
increases by the amount of 1 kilojoule. We can therefore now obtain a 
clear quantitative picture of the distribution of the geopotential unit 
layers. The bottom layer has the mean sea level as its base, and its 
thickness is approximately 1 decimeter. The layer has its maximum 
thickness at the 'equator (h^ = 1/9.78 meter 1.022 dm), and becomes 
gradually thinner toward the pole where it has its minimum thickness 
(h^ = 1/9.83 meter 1.017 dm). The next layer follows on top of this, 
with practically the same thickness as the bottom layer in each latitude, 
and so on. For practical estimates it is sufficient to remember that the 
thickness of the geopotential unit layer is roughly 2% thicker than 
1 decimeter and that its equatorial thickness is \% larger than its polar 
thickness. | 

Since the field of gravity is constant, the level surfaces are fixed sur- 
faces in space, and can therefore be used as reference surfaces indicating 
the vertical distance between a point in space and sea level. This dis- 
tance is uniquely determined by the value of the geopotential at the 
point, i.e., by the number of geopotential unit layers between the point 
and sea level. It is for this purpose convenient to introduce a termi- 
nology which is more suggestive of height than the energy expression 
kilojoule per ton. Since now the unit increase of geopotential roughly 
corresponds to a height of 1 decimeter, we are led to introduce for the geo- 
potential unit the name dynamic decimeter (abbreviated dyn dm). We 
therefore have by definition 

(1) 1 kj t" 1 - 1 dyn dm. 



95 Section 4- 12 

With the same terminology the geopotential of a unit mass at a point in 
space is referred to as the dynamic height of the point. When, for in- 
stance, a point has the dynamic height <p = 100 dyn dm, this means both 
that its vertical distance above the ground is roughly 102 dm, and that 
the potential energy of 1 ton at this point is 100 kj. The dynamic 
decimeter has of course the dimensions [L 2 T~ 2 ] of specific energy. 

A measure in frequent use is the dynamic meter (abbreviated dyn m), 
defined by 
(2) 1 dyn m 10 dyn dm. 

The dynamic meter is thus approximately 2% longer than the true 
meter. 

4-12. Analytical expression for the force of gravity. The expression 
for the force of gravity is derived from its relation to the geopotential. 
We return to 4-10 (4), which defines the concept of geopotential. This 
equation has, from 4-07(5), the following alternative forms: 

(1) -50 = g x bx + gy&y + gzdz = g-5r. 

The geopotential is according to 4-10(3) a single-valued function. Its 
variation from the point (x,y,z) to the point (x -f dx,y -f dy,z + 6z) is 

d0 d0 d0 

(2) d<t> - ^ dx + -~ 5y + -^ 8z = 5r-V0, 

d# by da; 

where 5r = i8x -f j&y + k&s. The expression in the middle is the scalar 
product of the vector 5r and the vector i(d0/d#) + j(d< /dy) + k(50/ds), 
whose rectangular components are the three partial derivatives of <t>. 
We introduce for this important vector the more convenient notation 
V< (read " del "). Hence by definition 

(3) v< =i ^ +j ^ + k ^. 

dx by cte 

This notation has already been introduced in the last expression on the 
right in (2). The expressions (1) and (2) are equal but opposite in sign. 
Hence, comparing the two scalar products, it follows that 

5rg - -6rV< or 6r(g -f V<) - 0. 

Since their scalar product is zero, one of the two vectors 5r and (g + V<) 
must be zero, or else they are perpendicular. The vector Sr is by defini- 
tion different from zero, and its direction is arbitrary. The only possi- 
bility is then g + V< 0, or 

(4) g = -V0. 

This is the analytical expression for the force of gravity. 



Section 4- 12 96 

The rectangular components of g are obtained by introducing in (4) 
the explicit form (3) of V<t>. Since the coordinate system is arbitrary the 
component of g along an arbitrary direction / is: 

d * 

(5) ft--^. 

In the standard system denned in section 4'03 the xy plane is tangential 
to the level surface. Therefore 

50 

(6) gx - 0; g y - 0; g z - - -g. 

oz 

The analytical expression (4) and the geometrical connection between 
the force of gravity and the geopotential unit layers (section 4-10) are 
completely interrelated, and the nature of this relationship is entirely 
mathematical. The physical meaning of the vector g and the function 
is purely incidental. Any vector which is analytically denned by one 
single scalar function through an expression like (4) is geometrically 
related to the unit layers of the function in the same way. A vector of 
this kind is called a potential vector, and the corresponding scalar function 
is called the potential of the vector. It will be shown presently that the 
pressure force is a vector of this kind. Owing to their fundamental 
importance in atmospheric dynamics, the geometrical properties of these 
vectors should be well known. We shall therefore derive these proper- 
ties once more without any reference to the special case of the field of 
gravity. 

4*13. Potential vector; ascendent; gradient. Let e=e(x,y,z) be 
any scalar function of the type discussed in section 4-04. This function 
may be represented by its unit layers, defined by the surfaces e = 0, 1, 2, 
. The vector Ve which has the function e as its potential is defined 
by an equation similar to 442(2): 



(1) 5s - 5r*Ve = |$r| |Ve| cos 0. 

By using the rules for the scalar product (section 4-07), we may derive 
the geometrical relation between the vector Ve and the unit layers of e. 
The displacement 5r is at our disposal and may be given an arbitrary 
direction in the field. If Sr is chosen perpendicular to Ve, their scalar 
product is zero. Hence from (1) 6e= 0; the displacement 5r is in a 
direction where the function e has no changes, i.e., in a surface e = const. 
The vector Ve is therefore directed normal to the equiscalar surfaces. To 
investigate its sense we shall give the displacement 5r the same direction 
as the vector Ve. Denoting the magnitude of this displacement with 



97 Section 4-14 

5w, it follows from (1), since cos 6 - 1 for this displacement, that 

(2) &-*|V|. 

The quantities on the right are the numerical values of the two vectors, 
and their product is therefore a positive quantity. The variation 6e 
in the direction of Vs is thus a positive quantity; in other words the 
vector Ve is directed toward increasing values of its potential. I ts magnitude 
as obtained from (2) is 



where h e denotes the thickness of the unit layer. The geometrical rela- 
tion between the vector Ve and the function e is thus identical to the rela- 
tion derived in section 4- 10 between gravity and geopotential, except 
for the sign. The vector Ve is known as the ascendent of the function e ; 
the opposite vector -Ve is called the gradient of e. The former is the 
mathematically convenient vector, whereas the latter generally is the 
physically important vector. With this new terminology the force of 
gravity is the geopotential gradient, which is a vector normal to the level 
surfaces, directed toward decreasing geopotential, and numerically equal 
to the reciprocal of the thickness of the geopotential unit layer. 

4*14. The pressure field in equilibrium. The force which balances 
the force of gravity and thus determines the state of equilibrium is the 
so-called pressure force, arising from the interaction between the air 
particles. It will be shown that there exists a relation between this force 
and the pressure field similar to that connecting the force of gravity and 
the geopotential field. 

To investigate the pressure field in equilibrium we shall apply the 
principle mentioned in section 4-08 to certain selected parts of the fluid, 
defined by closed boundaries, which in each case will be specified. In 
all cases the pressure forces in the interior of the selected part will appear 
in pairs, and hence give no contribution to the resultant. The only 
pressure forces left are those acting on the surface which encloses the fluid 
part. If the pressure at a certain point in this surface is />, the pressure 
force on an infinitesimal element 5-4 of the surface has, according to the 
definition 1-03(13), the magnitude pdA and is directed normal to the 
surface element. As stated in 2*04, the pressure at a point is inde- 
pendent of the orientation of the surface element on which it acts. 
Therefore, although defined as a force per unit area, it can at each point 
in space be expressed by a single numerical value. Thus pressure is a 
scalar quantity. 



Section 4- 14 



98 



To find the distribution of pressure, we apply the equilibrium condi- 
tion to a thin cylindric fluid element situated with its axis in a level 
surface (fig. 4'14a). The end faces of the cylinder are normal cross 
sections with area dA . Since the resultant of the forces acting upon this 






FIG. 4-14a. 



fluid element is zero, its component along the axis of the cylinder is also 
zero. The force of gravity has no component in this direction, and 
neither have the pressure forces acting on the side wall. There remain 
only the pressure forces on the two end faces. Hence 

(6) * pidA - p 2 5A = or pi = p 2 . 

This holds regardless of the length or orientation of the cylinder; conse- 
quently any equipotential surface is also a surface p = const, an isobaric 
surface. 

In order to study the variation in pressure from level to level, we con- 
sider a cylindric fluid element whose axis is normal to the levels, and 
whose base and top are contained in two levels with the respec- 
tive geopotential values <f> and <+&/> (fig. 4-146). It has just been 
shown that these levels also are isobaric surfaces defining certain definite 
pressure values p and p -f dp. We shall apply the equilibrium principle 
to this cylindric element and find the component of the resultant force 

along the axis. Let the mass of 
the cylinder be 8M. The force 
of gravity contributes to this 
component with its full amount 
gdM. The pressure forces acting 
upon the side wall of the cylinder 
have no component along the 
axis, whereas the pressure forces 




p 



FIG. 4- 



its cross-section area. 



on the top and bottom faces con- 
tribute with their full amount. 
Let 52 be the height of the cyl- 
Its volume dV and its mass 8M 



inder and dA 
are then 

(7) 

Using the expression 4-10(8) for g and choosing the positive direction 
along the axis upward, we have for the component of the resultant force 



99 Section 4- 15 

along this direction 



The equilibrium principle requires that the resultant be zero. Thus, 
when we introduce 8 A = 8V '/8z in the last term, we have 

(8) 

This equation can be used directly to discuss the distribution of pressure 
and mass in equilibrium. Before this discussion is taken up, the equa- 
tion will be used to derive the analytical expression for the resultant 
pressure force upon the cylindric element. 

4-15. The pressure gradient. The first term on the left in 4-14(8) 
is the total resultant of gravity acting upon the cylindric element. The 
total resultant of gravity is balanced by the total resultant of the pres- 
sure forces upon the boundary enclosing the element. It follows there- 
fore that the resultant pressure force is directed upward, so the pressure 
on the bottom face of the cylinder is greater than the pressure on the top 
face. In other words 8p is a negative quantity, and the pressure 
decreases upward. The resultant pressure force is thus a vector normal 
to the isobaric surfaces and directed toward decreasing pressure. Its 
magnitude is given by the second term on the left in 4-14(8). Since this 
is the pressure force on an element which has the volume 5V, the magni- 
tude of the pressure force per unit volume is 

8p 

> i 

where h p denotes the thickness of the isobaric unit layer. According to 
the discussion in section 4-13 the pressure force per unit volume is there- 
fore a potential vector, whose potential is the pressure. Since it is 
directed toward decreasing pressure, the pressure force per unit volume is 
the gradient of the pressure, or simply the pressure gradient. Using the 
notation of section 4*13 we can therefore write: 

(2) the pressure force per unit volume = -V. 

The expression (2) for the pressure force has been derived here for the 
state of equilibrium. However, this expression is completely determined 
by the pressure field and is thus entirely independent of the field of 
gravity, or of the simple relation between the two fields in the state of 
equilibrium. We may thus conclude that the expression (2) for the 



Section 4-15 100 

pressure force is valid for any state of motion of the fluid. This state- 
ment can be proved rigorously by more advanced mathematical meth- 
ods. We will, however, accept it here on the strength of the physical 
evidence and use the expression (2) when the equation of motion is 
derived. 

The rectangular components of the pressure force are obtained by 
introducing in (2) the explicit form of V/> from 4*12(3) : 

d/> dp bp 

(3) _v #= _i-?_j-?_k-f 

d* dy dz 

Since the system (x,y,z) is arbitrary, we conclude that the component of 
Vp along an arbitrary direction / defined by the unit vector 1 is 

(4) (_v/>), - -l-V/> - -|. 

In practical problems we shall generally want to distinguish between the 
vertical component - (d/>/ds)k and the horizontal component 



(5) 

of the pressure gradient. We have accordingly 

d/> 

(6) -Vp= -V//p-k-~. 

oz 

Since the vector -Vp is normal to the isobaric surface, the horizontal 
vector -V///> is normal to the lines of intersection between the isobaric 
surfaces and the level. These lines are the horizontal isobars in the 
constant-level maps. If dnji is an infinitesimal horizontal distance nor- 
mal to the isobars, and h p n is the normal distance between unit isobars, 
it follows from (4) 



(7) | v ,,p| 

Thus the two-dimensional relation (7) between the horizontal pressure 
gradient and the isobaric unit channels in a level surface corresponds to 
the three-dimensional relation (1) between the pressure gradient and the 
isobaric unit layers. In mts units (7) has the following meaning: If the 
normal distance between two isobars with a pressure difference of 1 cb 
(10 mb) is h p H ni, the horizontal pressure force per cubic meter is 
1/hpH mts units of force. 

4-16. The hydrostatic equation. From the expressions for the two 
forces which act upon the atmospheric elements the equation of relative 



101 Section 4- 16 

equilibrium can be written down directly. Consider an infinitesimal 
element of air which has the mass dM and the volume 8 V. From 4-12 (4) 
the force of gravity on the element dM is -6MV0. From 4-15(2) the 
force of pressure per unit volume is - V/>, and therefore the pressure force 
on the volume 5V is -5FV/>. In equilibrium the resultant of these 
forces is zero; hence 

(1) 0= -SMV</>-3FV/>. 

This equation may be referred to unit mass by dividing it by dM , or to 
unit volume by dividing by d V. Thus we obtain the equation of relative 
equilibrium in two equivalent forms 

(2) -- V0-V/>; 0= -pV<-V/>. 

These equations are valid for every air element in a resting atmosphere. 
For the further discussion of the hydrostatic state the vector equations 
(2) are most conveniently transformed into scalar equations. For this 
purpose we consider an arbitrary infinitesimal displacement Sr. The 
corresponding variations of geopotential and pressure are, according to 
4-13(1), 



Thus by performing scalar multiplication of the two vectors in (2) by 
the line element 6r we obtain 

(3) 6</>=-aty; dp=- P 8<t>. 

The second equation in (3) could incidentally have been derived directly 
from 4*14(8) by dividing out the two common factors d V and 8z. It will 
be shown presently that either of the equations (3) gives a complete 
dynamical description of equilibrium, and therefore both are equivalent 
forms of what is known as the hydrostatic equation. 

It was shown in section 4-14 that the isobaric surfaces coincide with the 
level surfaces in equilibrium. This fact is expressed mathematically by 
the equations (2), since the two gradient vectors are normal to their 
respective equiscalar surfaces. A geopotential layer of the dynamic 
thickness 8<t> will thus define an isobaric layer through which the pressure 
varies by the amount dp, so the variations 5<t> and 8p are characteristic 
for the whole layer. It follows then from (3) that the mass variables 

8<t> 8p 

(4) a-- and /> = --- 

Sp 50 

are constants throughout the layer. If the layer is of finite thickness, 
these equations define certain mean values of a. and p within the layer. 
For an infinitesimal layer ($< -> 0) the equations (4) give the values of a. 



Section 4-16 102 

and p on the level surface, and these values are constant on the level. 
Thus in the state of equilibrium the isosteric and the isopycnic surfaces 
also coincide with the level surfaces, and we can formulate the fol- 
lowing important rule: The state of equilibrium is characterized by 
complete coincidence between the surfaces of constant pressure and constant 
mass, both sets of surfaces being horizontal throughout the atmosphere. 

After the discussion of motion is taken up, it will be shown that certain 
states of motion are also characterized by the coincidence between the 
surfaces of constant pressure and constant mass. (However, only in 
the hydrostatic state are they also horizontal.) Whenever the state of 
the atmosphere is such that the isosteric surfaces coincide with the 
isobaric surfaces, the mass field is said to be barotropic (meaning: 
directed in accordance with the pressure field). In general the terminol- 
ogy is further simplified by omitting " the mass field," and thus the 
atmosphere itself is said to be barotropic when the above condition exists. 
In the usual case, where the isosteric surfaces intersect the isobaric sur- 
faces, the atmosphere is said to be baroclinic. With this terminology we 
have the rule : In the state of equilibrium the atmosphere is barotropic. 

The equations (2) can be given a simple geometrical interpretation by 
means of the unit layers, here given only for the first equation. Both 
vectors are numerically equal to the reciprocal of the thickness of their 
respective unit layers, by 4-13(3), so that 



(5) = a f J , or h p - ah^ 

Thus the isobaric unit layer contains a. geopotential unit layers. Explic- 
itly in mts units this rule may be expressed as follows: An isobaric layer 
of 1 centibar has a dynamic thickness of a dynamic decimeters. (If the 
variation of specific volume through the layer is appreciable, the mean 
value of a in the layer must be taken.) 

To get a rough idea of the order of magnitude of this thickness, con- 
sider a layer of dry air next to the ground, whose temperature at the 
100-cb level is 273K. The corresponding value of a. from the equation 
of state 2-09(1) is 287 273/100 = 784 m 3 t"" 1 . If the specific volume 
has this constant value throughout the isobaric unit layer 100-99 cb, the 
thickness of this layer is 784 dyn dm or approximately 800 dm. This 
gives the approximate rule that near the ground an increase in height of 
80 m corresponds to a pressure drop of about 10 mb, or 8 m to a drop of 
1mb. 

By a homogeneous atmosphere is meant an atmosphere of constant 
density (and hence constant specific volume). If the above value 
a - 784 m 3 tT 1 prevailed throughout the atmosphere, the pressure at the 



103 Section 4- 17 

height of 8000 m would be 1000 mb less than at the ground, or practically 
zero. Thus the height of a homogeneous atmosphere whose pressure 
and temperature at sea level are 100 cb, 273K is approximately 8000 m. 

4-17. Distribution of pressure and mass in equilibrium. It was 

shown in the preceding section that the mass field is barotropic in equilib- 
rium, and that the equiscalar surfaces of mass and pressure are both 
horizontal. The distribution of pressure and mass is known throughout 
the atmosphere if it is known along one single vertical. The distribution 
along a vertical is obtained by integration of the equations 4-16(3), 
yielding 

b & 



(1) 



,-*,-- [asp; p b -p a =- I 



The second of the equations (1) has mainly theoretical interest. It 
reveals the physical cause or origin of atmospheric pressure. By 4-10(5) 
we substitute 5</> = g8z and have 

b 



(2) 






Now pgdz is the weight of an air column of unit cross section and height 
dz. Thus (2) says that the pressure difference between any two levels a 
and b is the weight J b a pgbz of the column of air of unit cross section extend- 
ing between the two levels. Hence the pressure at any level is the total 
weight of a vertical column of unit cross section extending from this level to 
the top of the atmosphere. This is a principle to remember in all meteoro- 
logical work. 

The first equation (1) is known as the barometric height formula. It 
solves the fundamental problem of hydrostatics referred to at the begin- 
ning of the chapter to determine the dynamic heights which corre- 
spond to the various pressure values along an aerological ascent. Most 
of the remainder of this chapter will be devoted to a discussion of this 
equation. 

The aerological sounding gives simultaneous values of pressure, 
temperature, and relative humidity. According to sections 3-21 and 
3-23, from these data we can obtain the mixing ratio w. Thus for every 
point along the ascent we have the values of p, T, w. The correspond- 
ing value of a is found from the equation of state for moist air 3-25(5): 

m 

(3) a 



Section 4- 17 104 

where T* is the virtual temperature 325(4), and Rd is the specific gas 
constant for dry air. When this expression for a is introduced in (1), 

the integrand becomes -R<iT*8p/p R d T*d(ln p). Then (1) takes 

the form 



o 

.*/ 



(4) &- = -Rd] T*8(\np), 

a 

which is the barometric height formula adapted for practical use. 

Several graphical methods can be used to evaluate the dynamic height 
from formula (4) when simultaneous values of pressure and virtual tem- 
perature are known along a vertical. Before discussing these methods 
it is useful to investigate the state of equilibrium when the virtual tem- 
perature is a linear function of the geopotential. This simple tempera- 
ture distribution is not only of great theoretical interest, but it also 
serves as a useful approximation of real atmospheric conditions. 

4-18. The atmosphere with constant lapse rate. It is evident from 
4-17(4) that in hydrostatic problems the moist air may be treated as 
though it were dry air by replacing the real temperature by the virtual 
temperature. (This is one of the great advantages of virtual tempera- 
ture.) We shall systematically use virtual temperature in this chapter. 
In the special case of a dry atmosphere the virtual temperature auto- 
matically becomes the real temperature. 

By combining 4-16(3) and 4-17(3) the hydrostatic equation takes the 
form 

(i) s=-i? d r*^. 

p 

To integrate this equation the distribution of virtual temperature must 
be known, either as a function of pressure or as a function of the dynamic 
height <t>. The integration is particularly simple when T* is a linear 
function of <t>. 

We define the lapse rate of virtual temperature 7* at any point in the 
atmosphere by the equation 

5T* 
(2) fff. 

Thus 7* has the dimensions [0L~ 2 T 2 ], and in mts units is measured in 
degrees per dynamic decimeter. To say that T* is a linear function of 
is equivalent to saying that 7* is a constant or that the atmosphere has a 
constant lapse rate (of virtual temperature). 
We shall now determine the relations between 7 1 *, p, and <t> when the 



105 Section 4- 19 

lapse rate has the constant value 7*. We shall assume for the present 
that 7* > 0, i.e., that temperature decreases with height. We denote 
the arbitrary initial conditions at sea level by the notations: 

(3) r*=r *; p = po; 0=0. 
Integration of (2) with the initial condition (3) gives 

(4) r*=r-7*0 or - !_:*. 

^0 ^0 

When d<t> is eliminated between (1) and (2), we get 



Integrating (5) with the initial condition (3), we have finally 

vflrfV 
(6) 

From (2) and (6) we obtain the following important rule: When, in the 
state of equilibrium, the virtual temperature is a linear function of the geo- 
potential with the constant lapse rate 7*, it is also proportional to the power 

of the pressure. 
Elimination of T*/T% between (4) and (6) gives 

I 'v* h 

(7) p 

Equations (4, 6, 7) summarize the relations between T*, p, and 0. We 
note from (4) and (7) that both p and T* vanish at the level 

(8) *i-^|- 

7* 

This level fa is called the dynamic height of the atmosphere of constant lapse 
rate. For < > fa there is no more air. We note that fa depends only on 
TO and 7*. 

When 7* is given certain particular values, we obtain three important 
special cases to be discussed in the next three sections. 

4 19. The homogeneous atmosphere. When 7* is given the special 
value 1/Rd, then R<TY* = 1, and the formula 4-18(5) becomes 

a) ' = s j 

If we differentiate logarithmically the equation of state 3-25(5), 



Section 4- 19 106 

pa - RjT*. we get 

y* p a 

Comparing (1) and (2), we see that in the present case 8a 0, or 
a const. Thus when 7* = l/12<j we are dealing with a homogeneous 
atmosphere (defined in 4-16). Conversely, for a homogeneous atmos- 
phere, 

(3) 7* = yh - - 0.00348 deg (dyn dm)" 1 , 

amounting to a temperature drop of about 35C per dynamic kilometer. 
For a homogeneous atmosphere the equations 4-18(4, 6, 7) simplify to 

(4) r*=r *-^ ; -,= p --, p- t L * 



The dynamic height fa of the homogeneous atmosphere is obtained from 
either (4) or 4-18(8): 

(5) fa 



(This formula could have been obtained immediately by integrating 
4*17(1) for a. = const, and using the equation of state.) In the special 
case r - 273K, fa - 78,351 dyn dm ( 8000 m, as shown at the end of 
4-16). 

In the real atmosphere the lapse rate never is as large as 7^, except 
possibly in thin layers near the ground. The reason is that such an 
atmosphere would be too unstable, as will be shown in chapter 5. Still, 
the homogeneous atmosphere is a useful concept for theoretical discus- 
sion. 

When 7* > l/Rd, the density increases with height, giving even 
stronger instability than for the isosteric mass distribution just con- 
sidered. When 7* decreases from l/Rd, one gradually approaches con- 
ditions similar to those generally encountered in the real atmosphere. 
Another ideal case is the following. 

4-20. The dry-adiabatic atmosphere. When 7* is given the special 
value l/Cpd, then R<r(* = Rd/Cpd *d, and 4-18(6) takes the form 



(i) r* = T 

By substituting this expression for T* in 3-28(1), we see that the virtual 



107 Section 4-21 

potential temperature 0* at all levels is given by 

(2) 0* = T* I ) - const. 

\Po/ 

Thus an atmosphere with 7* = l/c p d is characterized by constant virtual 
potential temperature, meaning that all points (T*,p) lie on the same dry 
adiabat. From this property such an atmosphere is called a dry-adia- 
batic atmosphere. It has the lapse rate 

(3) 7* - Id - - 0.000996 deg (dyn dm)- 1 , 

Cpd 

amounting to a temperature drop of about 10C per dynamic kilometer. 
For a dry-adiabatic atmosphere, the equations 418(4, 6, 7) take the 
form: 

* & r* f P\** 

(4) r* = Tn -^ : * = I I ; t> = i 

\ / \J " rr*if \ J\ I 

Cpd * \fOf 

The dynamic height fa of the dry-adiabatic atmosphere is obtained from 
either (4) or 4-18 (8): 

(5) <bd = CpdT^. 

In the special case T* - 273K, fa - 274,100 dyn dm, or almost 
28,000 m. 

When in the dry-adiabatic atmosphere the virtual temperatures at the 
levels fa and fa are respectively T* and T*, we have from the first 
equation (4) : 

f/Z\ i I f T& T^\ < f\f\A / T^* 'T^* \ 

(o; 06 - fa = c p d(l a - 1 b ) = 1004(7 - Ib). 

It will be shown in 4-24 how this formula permits a quick evaluation of 
height when little accuracy is required. 

The transition from the homogeneous to the dry-adiabatic atmosphere 
corresponds to the decrease of 7* from approximately 0.0035 to approxi- 
mately 0.0010, with a corresponding increase of the top of the atmosphere 
from near 8 km to nearly 28 km. When 7* is decreased to 0, it is evident 
from 448(8) that the height of the atmosphere becomes infinite. This 
limiting case is discussed next. 

4 '21. The isothermal atmosphere. When 7* - 0, we have immedi- 
ately from 4-18(2) that T* has the constant value TQ throughout, so the 
atmosphere is isothermal. In this case the integrations of 4-18 no longer 
remain valid. However, we can in this case integrate 417(4) directly, 
obtaining 



(1) *- 

Po 



Section 4-21 108 

Solving for p, we get 

(2) />- 

Thus the pressure decreases exponentially upward, but never vanishes. 
This verifies the statement that the upper limit of the isothermal atmos- 
phere is at infinity. 

When in the atmosphere the virtual temperature between the levels a 
and 0& has the constant value T*, we have from (1) that 

(3) 0&-0 = # d r*ln^. 

d m n p b ' 

It will be shown in 4-24 how this formula is used for the accurate evalua- 
tion of height. 

The real atmospheric distribution of pressure and virtual temperature 
is rather complex and variable. However, within a thin layer the lapse 
rate may always be treated with good approximation as constant. Such 
a thin layer may therefore be considered as part of an atmosphere of con- 
stant lapse rate. 

4*22. Atmospheric soundings. An aerological sounding provides, as 
stated in 4-17, a set of simultaneous values of T* and p. These values 
can be plotted on any thermodynamic diagram of chapter 2. The points 
are customarily joined with straight lines, and the result is a polygonal 
curve representing a vertical column of the atmosphere. The curve is 
briefly called the (virtual temperature) sounding. 

The sounding should be distinguished from the process curves so far 
discussed on the diagram (for example, the adiabats). The process 
curves represent the change in the physical properties of one parcel of air 
during a process in which the parcel changes its state. These were much 
discussed in chapter 2. The sounding is an instantaneous picture of the 
state of a whole geometrical column of air. The distinction is analogous 
to that made in section 4-10 between the differentials d and d. A small 
variation of pressure on a sounding would be dp ; a small change of pres- 
sure during a process would be dp. 

4*23. Graphical representation of dynamic height. We shall now 
return to the practical problem of evaluating the dynamic height when a 
sounding is presented. The theoretical basis for this evaluation is the 
barometric height formula 44 7 (4), which we repeat here in slightly 
modified form : 

(1) 06 - 0a - R 



109 



Section 4-24 



This has a simple graphical representation on the emagram. Let the 
virtual temperature sounding between the levels a and b be plotted on an 
emagram. Let A denote the area bounded by the sounding, the isobar 
p - p a , the isobar p - p b , and the isotherm T = 0K. See fig. 4-23. 
Since the emagram is a graph of -In p against T, the area A is given by 

6 
(2) 



A = r*8(-ln />), 



just as an analogous formula, 2-32(3), held for the process curves of 
chapter 2. (The boundary of A in (2) is not a process curve, and the 
area A is not to be thought of as work or heat.) Combining (1) with 
(2), we see that <fo - <t> a = RaA. Thus the dynamic thickness of any 
layer in the atmosphere is proportional to the area A of fig. 4-23. 

Any diagram with the property (iii) of 2-32 is (up to a factor of pro- 
portionality) an area-preserving 
transformation of the emagram. 
This includes the tephigram, 
the (a,-)-diagram, and certain 
other diagrams not however 
including the Stiive diagram. 
Hence for any of these included 
diagrams we have the following 
rule: The dynamic height between 
the pressure levels a and b is (up to 
a proportionality factor fixed by the 
diagram) equal to the area enclosed 




300 



FIG. 4-23. Dynamic height on emagram. 
at the isobar p p b , 



by the virtual temperature sounding, the isobar p 
and the isotherm T = 0. 

Fig. 4-23 demonstrates how the dynamic thickness depends on the 
sounding. The warmer the sounding, the thicker must be a given iso- 
baric layer, since the sounding encloses a larger area. This is physically 
obvious. The warmer air is less dense, and it takes a longer unit column 
between levels a and b to build up the weight of air which, according to 
4-17, is responsible for the given pressure difference p a - p b . The same 
fact may be seen from 4-16(5), according to which the thickness of an 
isobaric layer is directly proportional to its specific volume. 

4*24. Adiabatic and isothermal layers. According to the rule of 
4-23, two different soundings between the pressure levels a and b will 
have the same dynamic thickness whenever they have equal areas A. 
To evaluate the thickness of an arbitrary layer, we replace the real 



Section 4-24 



110 



sounding between a and & by a fictitious sounding which has an equal 
area. We give the fictitious sounding such a temperature distribution 
that its dynamic thickness is easy to evaluate. In practice either dry- 
adiabatic or isothermal layers are chosen, since their evaluation was 
shown in 4-20 and 4-21 to be simple. Both give useful methods for the 
evaluation of height. 

a. Determination of height by dry-adiabatic layers. The real virtual 
temperature sounding between the levels a and b is represented by the 
heavy curve in fig. 4-24<z. It defines an area A like that shaded in 
fig. 423. Each adiabat between the levels a and b defines a certain area 
A ' of the same type. In order to make A ' equal A , we choose the adia- 
bat for which the areas A \ and A 2 in fig. 424a are equal. This is easily 
estimated by the eye. Then the fictitious dry-adiabatic layer between 
the levels a and b is equivalent in dynamic thickness to the real layer. 

According to 4-20(6), the dynamic thickness of the dry-adiabatic layer 
is given by 



(1) 



- <t> a - 1004(7? - 7?) (in dyn dm). 



Here 7? and 7? are the virtual temperatures where the dry adiabat 
crosses the isobars p = p a and p = pi,, respectively. See fig. 4-24a. In 




FIG. 4-24a. Equivalent adiabatic 
layer. 



FIG. 4-246. Equivalent isothermal 
layer. 



practical work heights are generally measured in dynamic meters. 
Using the symbol // for dynamic height measured in dynamic meters, we 
have the following approximate formula for the dynamic thickness: 



(2) 



// 6 - II a 100(7? - 7?) (in dyn m). 



Formula (2), first suggested by Stiive, is very convenient for rapid work 
when only approximate values are needed. Each 0.1C error in reading 
7? or 7? yields an error of 10 dyn m in H b - H a . For accurate height 



Ill Section 4- 25 

evaluation a second method based on the isothermal layer is therefore 
needed. 

b. Determination of height by isothermal layers. The real virtual 
temperature sounding between the levels a and b is represented by the 
heavy curve in fig. 4-24&. By visual estimation a mean isotherm 
7"= 7^ is drawn so that A% = A 4 . By the construction the fictitious 
isothermal layer T = T^ has the same area A as the real layer. Hence 
by the rule of 4-23 this fictitious layer has the same dynamic thickness 
as the real layer. But, according to 4-21(3), the dynamic thickness of 
the isothermal layer is given by 

* Pa 

For practical purposes we prefer to express the mean virtual tempera- 
ture in centigrade and the height in dynamic meters. Then (3) becomes: 

(4) Ih - Ua - 28.7(273 + ) In > 

Pb 

where ^ is the mean virtual temperature in degrees centigrade. 

Formula (4) is the basis of all accurate height computations. For 
routine work tables have been constructed from (4), based on the tables 
of V. Bjerknes (1912). 

4-25. The Bjerknes hydrostatic tables. Since the three variables 
C> Pa, and pb in formula 4-24(4) vary within wide limits, no direct tabu- 
lation of lib Ha is possible in one single table. Bjerknes invented four 
tables* which conveniently solve all problems of determining dynamic 
height. In the tables pressures are expressed in millibars. The tables 
are referred to by their original numbers to facilitate reference. In 
section 4-26 we shall discuss the U.S. Weather Bureau's adaptation of 
these tables. 

The standard isobaric surfaces are the surfaces for which p = 1000, 900, 
, 100 mb. The standard isobaric layers are the layers between adja- 
cent standard isobaric surfaces, for example, between 800 mb and 700mb. 

Table 9M is computed directly from 4-24(4) by giving p a /pb succes- 
sively the values 10/9, 9/8, -, 2/1, and C integral values in various 
ranges between -109C and 49C. For each value of p a /pb and each 
tabulated value of /* , the table gives H b - H a in dynamic meters. Hav- 
ing obtained C fr m an emagram for a standard isobaric layer, table 9M 
gives directly the dynamic thickness of the layer. The part of table 9M 
covering the layer between 1000 mb and 900 mb is given here. 

* V. Bjerknes, Dynamic Meteorology and Hydrography, Washington, 1912. 



Section 4-25 



112 



TABLE 9M 
DISTANCE BETWEEN STANDARD ISOBARIC SURFACES IN DYNAMIC METERS 



900mb 



1000 mb 



4 





1 


2 


3 


4 


5 


6 


7 


8 


9 


-40 


705 


702 


699 


696 


693 


690 


687 


684 


680 


677 


-30 


735 


732 


729 


726 


723 


720 


717 


714 


711 


708 


-20 


765 


762 


759 


756 


753 


750 


747 


744 


741 


738 


-10 


795 


792 


789 


786 


783 


780 


777 


774 


771 


768 


- 


826 


823 


820 


817 


814 


811 


808 


804 


801 


798 


+ o 


826 


829 


832 


835 


838 


841 


844 


847 


850 


853 


10 


856 


859 


862 


865 


868 


871 


874 


877 


880 


883 


20 


886 


889 


892 


895 


898 


901 


904 


907 


910 


913 


30 


916 


919 


922 


925 


928 


931 


935 


938 


941 


944 


40 


947 


950 


953 


956 


959 


962 


965 


968 


971 


974 



The derivation of the other three tables is best understood after a 
simple transformation of the fundamental formula 4-24(4). This for- 
mula can be expanded into two terms. The first term is 



(1) 



(28.7) (273) In i 
Pb 



which is the dynamic thickness of the layer when 
term may be written 



(2) 



A/7. 



t - 0C. The second 



When these two 



which becomes a temperature correction to (1). 
expressions are introduced into 4-24(4), we have 

(3) ffft-fla- (//6-//a)o+A//. 

Thus when (7/6 - // a )o and A// are obtained separately, the true dy- 
namic thickness //& - H a is obtained by algebraic addition. 

Table 10 M is computed from (1) by giving p a successively the stand- 
ard pressure values 1000, 900, , 100 mb. For each of those values, 
pb is given pressure values for each millibar from 100 mb above p a to 
100 mb below p a > For each value of p a and />&, table 10M gives the 
dynamic thickness (Hb - // a )o when / = 0C. A sample of the part of 
table 10M for p a = 1000 mb is given here, covering pb between 1000 mb 
and 1049 mb. All distances are negative, since p b > p a - 

Table 11M isf computed from (1) by giving p a the value 1100 mb, a 
value chosen to avoid negative heights. The pressure pb is here ex- 

t Called table 1 1 *M in the original tables. The orginal table 1 1M is a seldom used 
combination of tables 9M and 10M, set up in different form. 



113 



Section 4*25 



TABLE 10M 
DISTANCES FROM THE 1000-MB SURFACE AT 0C IN DYNAMIC METERS 



1000mb. 



Pressure 
(mb) 


01234 


56789 


1000 
1010 
1020 
1030 
1040 


-8 -16 -23 -31 
-78 -86 -93 -101 -109 
-155 -163 -171 -178 -186 
-232 -239 -247 -254 -262 
-307 -315 -322 -330 -337 


-39 -47 -55 -62 -70 
-117 -124 -132 -140 -147 
-193 -201 -209 -216 -224 
-270 -277 -285 -292 -300 
-345 -352 -360 -367 -375 



tended over all values from 1099 mb to 1 nib, and the table gives the 
dynamic thickness (lib // )o of a layer between p a and pb when 
/* = 0C. By using this table for any two pressure levels pb and p bt 
we can obtain by subtraction the dynamic thickness of a 0C layer 
between the two levels pb and p b . As table 11M is seldom used in prac- 
tice, no sample of this table is given. 

Table 12M gives the temperature correction AH which is applied to all 
heights obtained from tables 10M and 11M. It is computed from 
formula (2) with (7/6 - H a ) G as one argument and t* n as the other. The 
correction has the same sign as /* . A sample of table 12M is given here 
for (H b H a )o between 150 and 190 dyn m and for * between and 
49: 

TABLE 12M 
TEMPERATURE CORRECTION FOR TABLES 10M AND 11M 



(lib H a )o 
(dyn m) 


fm\ <-; 





10 


20 


30 


40 


1 2 


3 


4 


5 


6 7 


8 9 


150 





5 


11 


16 


22 


1 1 


2 


2 


3 


3 4 


4 5 


160 





6 


12 


18 


23 


1 1 


2 


2 


3 


4 4 


5 5 


170 





6 


12 


19 


25 


1 1 


2 


2 


3 


4 4 


5 6 


180 





7 


13 


20 


26 


1 1 


2 


3 


3 


4 5 


5 6 


190 





7 


14 


21 


28 


1 1 


2 


3 


3 


4 5 


6 6 



The temperature correction A// for (//ft # a )o =173 dyn m and / = 34 is calculated as follows: 
For 170 dyn m and 30C, we read 19 dyn m. For 170 dyn m and 4C, we read 2 dyn m. Hence 
AH = 19 + 2 = 21 dyn m. 

Tables 10M and 12M, according to (3), determine the height above or 
below the nearest standard surface of any salient point of the sounding 
curve (highest point, tropopause, inversion, front, etc.)- Tables 11M 
and 12M determine by double operation and a subtraction the height 
between any two salient points, for example, between the surface of the 
earth and the top of the ascent. Table 11M should only be used for 
rather rough estimates of the total height, since appreciable errors may 



Section 4- 25 114 

be brought in by the graphical estimate of the mean virtual temperature 
of a thick layer. 

Greater accuracy is obtained when the total layer is divided into the 
standard layers, and its thickness measured as the sum of the thicknesses 
of these layers. Thus the accurate determination of height by means of 
the tables is performed by the following three operations : 

1 . Determination of the dynamic height of the 1000-mb surface. This 
is done with the aid of table 10M and the correction table 12M. 

2. Determination of the dynamic thicknesses of each of the standard 
layers. These are obtained directly from table 9M. 

3. Determination of the dynamic height of the top of the ascent above 
the highest standard isobaric surface. This is done with the aid of 
table 10M and the correction table 12M. 

When the results of 1, 2, and 3 are added we obtain the total dynamic 
height of the sounding, and also the dynamic heights of the standard iso- 
baric surfaces. The heights of other salient points along the ascent 
curve can be found from tables 10M and 12M. These points are, how- 
ever, determined more conveniently and with sufficient accuracy through 
an interpolation on a pressure dynamic height curve. This curve can be 
drawn on the same emagram where the original virtual temperature 
sounding curve was plotted for the determination of the values of t^ 
(see fig. 4-27). It is constructed from the dynamic heights of the stand- 
ard isobaric surfaces. We use the pressure scales of the emagram and 
introduce a horizontal scale of dynamic height, increasing from right to 
left, with at sea level. In an isothermal layer the pressure dynamic 
height curve is a straight line, according to 4-21(1). Elsewhere it is 
slightly and smoothly curved. The lack of smoothness of the curve 
indicates errors in the height computation. The height of any salient 
point is found by following the isobar from the sounding curve to the 
pressure dynamic height curve, and reading the corresponding dynamic 
height on this curve. 

4-26. U.S. Weather Bureau hydrostatic tables. The U.S. Weather 
Bureau has published hydrostatic tables used in all official height evalua- 
tions in this country. They are computed from Bjerknes' tables, but 
they are different in two respects : First, all heights are expressed in terms 
of the unit 0.98 dyn m. This is a unit nearly equivalent to the geometric 
meter. All dynamic height values in these tables are thus 2% larger 
than those in Bjerknes' original tables. Second, there have been added 
a number of interpolated standard isobaric surfaces for greater accuracy, 
namely, the 350, 250, 175, 150, 125, 80, 60, 50, 40, 30, 20, 15, 10, and 
5-mb surfaces. Having noted these general differences, we describe 
these tables briefly: 



115 Section 4-27 

Table 1 is Bjerknes' table 9M, with some unimportant reductions in 
the tabulated range of /. 

Tables 2 and 2a are Bjerknes' table 10M. Except at pressures above 
800 mb the arbitrary pressure levels pb are referred only to the next higher 
standard pressure p a > pi,. 

Table 3 is Bjerknes' table 12M, made longer and simpler by tabulating 
directly for every degree of C- 

There is no table corresponding to Bjerknes 1 table 11M. 

4-27. Height evaluation on a diagram. An accurate graphical 
method for height determination has been invented by Vaisala. It com- 
bines the emagrarn with certain scales which are equivalent to Bjerknes' 
tables. The scales will be described in the same order as the tables in 
4-25, in order to facilitate comparison. An emagram with Vaisala's 
scales is drawn in fig. 4-27. The dry adiabats and other curves are 
omitted for clearer reading. 

1. The first scales to observe are those located along isobars in the 
middle of each standard layer. These scales correspond to table 9M, 
from which they have been plotted, and on a large-scale diagram can 
be read to an accuracy of nearly one dynamic meter. To get the thick- 
ness in dynamic meters of any standard isobaric layer, we simply read 
this scale at the point where the mean virtual temperature isotherm 
crosses this scale. 

2. The double scale labeled B at the bottom of the diagram gives the 
dynamic thickness (//iooo ~ H P )Q of the layer between the 1000-mb 
level and the pressure p. It assumes the mean virtual temperature 0C. 
This replaces the most commonly used part of table 10M. It has an 
accuracy of about one dynamic meter on a larger diagram showing 
more detailed scales. It is used to get the dynamic height of the 
1000-mb level above or below the station for stations not too far from 
sea level. 

3. A scale on the right edge of the diagram gives directly the dy- 
namic height in dynamic kilometers from the 1000-mb level to any other 
pressure level p, when the mean virtual temperature of the column is 0C. 
This scale replaces table 11M, except that Vaisala has used 1000 mb 
instead of 1100 mb as the reference level. By means of this scale and 
a subtraction the dynamic thickness may be determined between two 
arbitrary pressure levels in an atmosphere where f^ = 0C. The accu- 
racy is about 10 dynamic meters on a detailed scale. 

4. The scale labeled A at the bottom of the diagram gives the factor k 
by which the heights derived in 2 and 3 above must be multiplied, when 
the mean virtual temperature is different from 0C. Scale A uses the 
temperature scale of the emagram. This temperature correction doeo 



Section 4-27 



116 



200 

300 

1400 
JS 
*500 

600 

700 

800 

900 

1000 

c 

A -< 


1 


3 \ 


-^dynkm AH B 
^765432 1,,,*,< 


\ 

12 
11 
10 
9 

| 

S"* 
4 
3 

2 
1 

A 


















675 


QLt.A* 


2500 2( 


oo y; 


002801 

V 


\ 

1800 


\ 






100 








m* 


*+> 

7L2L- 


1900 


201 
\ 


\ '' 


'1400* 


' ' V 


\ 
\ 

m ' 


\ , 

1600 


\ 


1700 ' 






1860 


-57tfl- 




'1200 






\ 


^ 

\ 


, , . 


1500 


u. 


LSJL9- 




1300 






1000 ' 





' 1100 





\ 

'120 


\ t \ 

i X V 


'i; 


bo 


-3120- 














\ 


\ 

L --.-> 


1082 


-80S8- 


wo 




100 




^ 
















V 


O/f / 

yojf, 
x \^ 


-wtfi- 


800 






900 




\ 
















J 


865 


S09^- 


7W 








JOO 






^ 


0*C -40* -20* 0* ^mnmmnmn^^^^m^rfir 

10* -40* -20* 0' 20* 40* t 



.80 .85 .90 .95 - 1.00 1.05 1.10 k 

-200 200 400 



940 960 980 1000 1020 1040 1060 p 

FIG. 4-27. Computation of height on emagram. 

the same job as table 12M, but is designed for slide rule computation. If 
the mean virtual temperature is /C, then the dynamic thickness 
Cff 6 - H a )t at /C is given by 

From 4-25(2, 3) we see that k - 1 + (//273). 



117 Section 4-28 

By the use of the scales described in 1, 2, and 4 we can determine the 
heights for a sounding beginning at any pressure within the range of 
scale B. This method is quite as accurate as the use of Bjerknes' tables. 
The errors in either method are less than the instrumental errors. For 
pressures below 200 mb, Vaisala's diagram contains a low-pressure scale 
(described in 2-32), but this has been omitted in fig. 4-27. There is 
customarily a scale of the type described in 1 above associated with these 
low pressures. 

4-28. Example of height computation on the diagram. We suppose 
that a radiosonde observation has been received for a station whose ele- 
vation is 50 m (49 dyn m) above sea level. By some procedure the 
virtual temperatures t* have been computed at the pressures p received 
in the ascent, and we have the following table for p and /*. The highest 
pressure is at the ground. 

TABLE 4-28 



1020 8 540 -3 

960 14 400 -20 

870 13 280 -40 

700 8 200 -50 

This virtual temperature sounding is plotted as a solid line in fig. 427, 
with the customary linear interpolation between the points received. 
The following steps are carried out in order. They have been performed 
on a larger diagram with more detailed scales than fig. 4-27, but the 
reader can follow the work on that figure. 

1. The dynamic height II 8 in dynamic meters of the station is written 
in the right-hand corner of the diagram : 49. 

2. The dynamic thickness (//iooo - ^)t of the layer up to 1000 mb 
is obtained as follows: Opposite the ground pressure 1020 mb on scale B 
is read (//iooo - H 8 ) = 155 dyn m. From the sounding we see that the 
mean virtual temperature for the layer 1020 mb to 1000 mb is 9C. 
Opposite 9C on scale A is read k = 1.033. According to equation 
4-27(1), (// 100 o -#*)* = (155) (1.033)= 160 dyn m. The number 
160 is written between 1020 mb and 1000 mb in a column to the left of 
the station elevation. 

The reader can obtain (ffiooo - #)o = 155 from our sample of table 
10M, with an additive correction AJ7 = 5 from our sample of table 12M. 
This gives (#1000 - #)< = 160, confirming the above graphical work. 

3. The dynamic height HIQOO of the 1000-mb level is obtained by 



Section 4-28 118 

adding 160 to 49, giving 209 dyn m; 209 is written above 49, on the 
1000-mb isobar. 

4. The dynamic thickness //goo - //iooo of the layer from 1000 mb 
to 900 mb is obtained as follows: A mean isotherm C IS found for this 
part of the sounding in fig. 4'246. (A transparent ruler is usually 
used for this.) In this case /* = 13C, and this isotherm crosses the 
height scale for this layer at 865 dyn m. This is the dynamic thickness 
o f the layer in question; it is posted in this layer above the figure 160. 

The reader can also obtain //goo - //iooo =* 865 from our sample of 
table 9M. 

5. 865 is added to 209 to give the dynamic height 1074 dyn m for the 
900-mb level. 

6. In similar fashion the dynamic thickness //& II a is found for 
each of the standard isobaric layers, and the values are posted in the same 
column above 160. 

7. The dynamic heights of the standard pressure levels are posted in 
the column above 49. They are always the cumulative totals of the 
lib H a for lower layers, plus the dynamic height of the ground. All 
these heights have been posted in fig. 4-27, ending with // = 12,121 dyn m 
at 200 mb. 

8. A pressure dynamic height curve is drawn as follows: The same 
pressure scale is used, but the horizontal axis is now converted into a 
dynamic height scale. It increases from right to left and is labeled in 
dynamic kilometers at the top of fig. 4-27. With these scales the ground 
pressure and each of the standard pressure values are plotted against the 
corresponding dynamic heights from the last column. For example: at 
p 1020 mb is plotted // = 49 dyn m ; at p = 1000 mb is plotted // = 209 
dyn m ; etc. In fig. 4-27 the last point plotted is 9446 dyn m at 300 mb. 
These points are drawn as circles in the figure. 

The points just plotted are joined with straight line segments to form 
the pressure dynamic height curve (the broken curve in fig. 4*27). 
This curve should show no marked irregularities; if such exist all the 
work should be checked. The curve should be more or less parallel to 
a dry adiabat (see fig. 2-32&). 

From the pressure dynamic height curve and the sounding one can 
obtain all necessary information about the ascent. For example, the 
pressure at 10,000 ft (2990 dyn m) can be read as the pressure opposite 
2.99 dyn km on the pressure dynamic height curve. It is 716 mb in this 
case, and the error should never exceed 1 mb. The virtual temperature 
at 10,000 ft can be read as the temperature where the sounding crosses 
the pressure 716 mb. It is 8.3C in this case, and the accuracy is limited 
only by the original data. 



119 Section 4-30 

4-29. Further remarks. The steps 2 and 3 of the method of 4-28 are 
modified slightly in case the ground pressure is less than 1000 mb. As 
long as the pressure is within the range of scale B in fig. 4-27, the idea is 
to introduce a fictitious isothermal column down to 1000 mb. The 
thickness -Hiooo Us of this fictitious column is obtained from scale B. 
This number will be negative, and it is therefore subtracted from the 
ground elevation to give the dynamic height of the fictitious 1000-mb 
level. By using the fictitious and the real columns, the thickness of the 
layer between 1000 mb and 900 mb is then determined, and the process of 
4'28 is continued. 

If the ground pressure is too low for scale 5, it is necessary to compute 
the dynamic height from the ground up to the next standard pressure 
level by use of 4-24(4), or from Bjerknes' tables. After that the method 
of 4-28 can be applied. 

The scale described in 3 of 4-27 is never used for accurate height evalu- 
ations; it is used only for rough estimates of height. 

Vaisala's diagram contains two other scales not mentioned so far and 
not shown in fig. 4-27. The first gives immediately the difference 
between the actual temperature t and the virtual temperature t* for 
saturated air. If the air is unsaturated the correction is mentally 
multiplied by r. With these scales the virtual temperature sounding 
may be plotted readily from the true temperature sounding. The other 
scale is designed to give directly the height correction in dynamic meters 
for saturated air due to the difference between /* and /. If these scales 
are used, the true temperature sounding is plotted instead of the virtual 
temperature sounding. The procedure of 4-28 is then carried out on the 
sounding as plotted. Before each height is posted, however, there is 
added to it the height correction for saturated air (from the scale) 
multiplied by r. This yields the same values of Hb H a as the method 
of 4'28. The two methods give equally accurate results, and the choice 
between them is a matter of taste. 

4-30. U.S. standard atmosphere. The U.S. standard atmosphere 
(also called the N.A.C.A. atmosphere) was adopted in 1925 by the 
N.A.C.A.* to serve as a standard for all aeronautic work. The standard 
atmosphere is conveniently described by letting the pressure p be the 
independent variable, and considering standard temperature T p and 
standard altitude z p as unique functions of p. These functions may be 
obtained by a pressure-height calculation on a fictitious column of dry 
air with the following properties: 

* National Advisory Committee for Aeronautics, Technical Reports 147, 218, 246, 
538, Supt. of Documents, Washington, D.C. 



Section 4-30 120 

(i) The level where the pressure is PQ (= 101.33 cb) serves as the 
origin for measuring standard altitude, and is called the base level. The 
base level either may or may not coincide with sea level. 

(ii) The standard temperature T p at the base level is 288K (15C). 
Up to the level where z p is 10,769 m, the standard temperature drops 
6.5C per (geometric) kilometer rise, corresponding to the constant 
lapse rate 

(1) 7 - deg (dyn dm)"" 1 . 

n 

At and above 10,769 m the standard temperature has the constant value 
218K (-55C). Thus 

(288 - 0.0065z p (z p ^ 10,769 m) 

(2) P ~l218 (z p ^ 10,769m). 

(iii) Gravity is assumed to have the standard value g n (= 9.80665 
m s~~ 2 ) of section 1-07, and Rj is assumed to be 287.076 kj t" 1 deg"" 1 . 

We can obtain p as a function of z p (below 10,769 m) by putting (1) 
into 4-18(7). We get 



to oofi^ 
1 - ^T 



P" Po 

Substituting the constants from (iii) and using (2), we get 

- 0.00002257z p ) 5 - 266 ] 
(3) p-{ /r p \ 5 - 250 \ (z p S 10,769 




From (3) the pressure p a at the base of the isothermal layer is 
Pa - po(0.75694) 5 ' 256 = 0.23145/>o - 23.452 cb. In the isothermal layer 
formula 4-24(3) yields the following: 

--. 

P 

Setting <t> - a = gn(z P - 10,769) and p a - 23.452, we can sofve for z p , 
obtaining 

23 452 

(4) Zp - 10,769 + 6381.6 In - (z p * 10,769 m). 

P 

To exhibit p as the independent variable, we solve (2) and (3) for 
T p and z p , and combine them with (4). Thus 

//A 0.19028 

288 (p^ 23.452 cb) 



r 
T p 




(p 23.452 cb). 



121 Section 4-31 

f T /\0.19028-| 

44,308 1 - ( ) (p ^ 23.452 cb) 

(6) 2p = L Vft)/ J 2 

10,769 + 6381.6 In ?^*52 ( p ^ 2 3.452 cb). 

L P 

Formulas (5) and (6) completely describe the standard atmosphere. 
They obscure the fact that this atmosphere has a constant lapse rate in 
its lower portions. On the other hand, they emphasize the fact that 
pressure is the independent variable to which both standard tempera- 
ture and standard altitude are referred. Based on these formulas, there 
have been constructed tables of the U.S. standard atmosphere which are 
convenient for use in computations. (See, for example, N.A.C.A. 
Technical Report 538.) 

Since, by (5), T p is a function of />, a sounding of the standard atmos- 
phere may be plotted on any thermodynamic diagram. As explained 
in section 4-23, the standard altitude z p is proportional to the area on an 
emagram to the left of the sounding of the standard atmosphere between 
the isobars p Q and p. The construction and study of such a diagram will 
answer most theoretical questions about the standard atmosphere and 
the pressure altimeter. 

It should be observed that the definition of the U.S. standard atmos- 
phere is not given in dynamic terms, since the lapse rate is originally 
expressed with geometrical meters instead of with dynamic height units. 
As a result formulas (5) and (6) are mutually consistent only when 
gravity has the standard value g n . If gravity has a constant value g 
other than g n , the reader may show that formula (6) is a correct expres- 
sion for altitude as a function of pressure in an atmosphere for which the 
temperature is equal to T p multiplied by g/g n - 

4-31. The pressure altimeter. In an airplane height is usually 
measured with a pressure altimeter. This instrument is an aneroid 
barometer whose dial is graduated in height units instead of pres- 
sure units. The scale is made with equal height units spaced uni- 
formly around the dial. At its normal setting the mechanism is cali- 
brated so that at each pressure p the pointers indicate the standard 
altitude given by 4-30(6). At all levels in the standard atmosphere the 
altimeter with this setting will correctly indicate the true altitude above 
the base level defined in 4-30 (i). 

In practice it is desired to have the altimeter indicate altitude above 
sea level (briefly called sea level altitude) . This is accomplished by rotat- 
ing the height scale relative to the barometer and the pointers. If the 
altimeter is in the standard atmosphere, the height scale is rotated just 



Section 4-31 122 

far enough to increase all height readings by the sea level altitude of the 
base level. After this is done, the altimeter will correctly indicate the sea 
level altitude of any level in the standard atmosphere. (In practice the 
height scale is fixed, and the rest of the mechanism is turned; the effect 
is identical with that just described.) 

Before the height scale was turned, the instrument indicated zero 
altitude at the pressure po (which is equal to 29.92 in. of mercury). In 
this position the " altimeter setting " is said to be 29.92. After the 
height scale was turned, the instrument indicated zero altitude at sea 
level. The pressure at which the instrument will read zero altitude is 
called the altimeter setting. It is usually expressed in inches of mercury, 
and in the standard atmosphere we have just seen that it is also the 
pressure at sea level. 

The altimeter setting is the only pressure indication which is visible 
on the altimeter. It serves at all times to represent the amount by 
which the height scale has been rotated from its normal setting. When 
the altimeter setting p\ is put in 4-30(6), it determines an altitude z Pl 
which may be positive or negative. No matter what the true atmos- 
phere is like, a mechanically perfect altimeter with the altimeter set- 
ting pi will indicate at each pressure p the corresponding standard alti- 
tude z p minus z Pl . 

In each airways weather station there is computed several times a day 
the altimeter setting which will cause an altimeter to indicate the true 
altitude of the field when the plane is resting on the runway. The reader 
may show from the last paragraph that the correct altimeter setting may 
be determined as follows: 

(i) Determine the standard altitude z p of the field by putting the 
pressure at the field into 4-30(6). 

(ii) Subtract the true altitude z of the field from z p , to obtain a 
height. Let this height be denoted by z Pl . 

(iii) Determine from 4-30(3) the pressure pi corresponding to the 
height z Pl . This pressure is the altimeter setting. These operations 
may be performed quickly with the aid of tables of the standard atmos- 
phere. In practice the whole procedure is incorporated into a single- 
entry table giving the altimeter setting p\ for each pressure p at the field. 

It should be stressed that the altimeter setting for any one station 
depends only on the station pressure. In this respect it is very different 
from the variable reduction to a " sea level pressure " used as the pres- 
sure report for the synoptic maps. 

4*32. Altimeter errors. Apart from mechanical and calibration 
errors a pressure altimeter is subjected to two types of errors which are 



123 Section 4-32 

inherent in its design. The first arises when a plane lands at an airport 
with an altimeter setting which is incorrect for that airport. The second 
arises when a plane is high over the ground, if the intervening atmosphere 
is different from the standard atmosphere. 

The first error is rather simple to visualize and measure. From 
430(6) each millibar variation in p represents (near /> ) a variation of 
between 8 and 9 m in z p . Thus an altimeter will be wrong by 8 or 9 m 
for each millibar error in altimeter setting (about 3 m for each 0.01-in. 
error in altimeter setting). For example, suppose an altimeter was set 
correctly for a sea level field of take-off where the pressure was 1030 mb. 
Then on landing with the original setting at a sea level field where the 
pressure is 980 nib, the plane will hit ground when the altimeter indicates 
420 m above sea level! 

To explain the second error suppose that an altimeter is set correctly 
for a sea level station under the plane. Suppose the mean virtual 
temperature T^ of the air column beneath the plane differs from the 
mean temperature TA of the standard atmosphere. Suppose also that 
gravity g in the column differs from the value g n of the standard atmos- 
phere. Then the altimeter will have an error which can be estimated 
from 4-24(3). Let the sea level pressure be p a \ let the plane be at the 
pressure pb. Then the true dynamic height of the plane is 



(1) 

Pb 

The altimeter will indicate an altitude corresponding at gravity g n to 
the dynamic height <J>A given by 

(2) 0A-287r A ln^. 

Pb 

Comparing (1) and (2), we see that 

T* 



Hence the true altitude z - </>/g is given in terms of the indicated altitude 
ZA - 4>A/gn by the formula 

rr\% rrt% 

,~\ * m f>n * m 

(3) a- -- ZA Z A . 

TA g T A 

Since TA is not far from 280K at the lower flight altitudes, we see from 
(3) that each 2.8C departure of T* from T A will cause about a 1% 
error in the altimeter. If the actual air is colder than the standard 
atmosphere, the altimeter will read too high. If the actual air is warmer 



Sect ion 4- 32 124 

than the standard atmosphere, the altimeter will read too low. Since g 
never differs from g n by more than J% at flight levels, the maximum 
gravity error is equivalent to the error introduced by a variation in T 
of merely 0. 7C. Thus the variation of gravity can usually be ignored in 
comparison with the variation of !T*. 

The determination of TA in practice can be made exactly from tables. 
The accurate determination of T% is very difficult in flight, since the 
plane is constantly passing over new columns with unknown tempera- 
ture distributions. 



CHAPTER FIVE 
STABILITY OF HYDROSTATIC EQUILIBRIUM 

5-01. The parcel method. In this chapter we shall develop methods 
for determining the vertical stability at any level of the atmosphere 
where hydrostatic equilibrium prevails. The level of the atmosphere 
which will be investigated is called the reference level. 

First imagine that a small parcel of air is displaced infinitesimally 
upward or downward from this reference level. By studying the subse- 
quent motion of the displaced parcel, we shall formulate certain criteria 
of stability: If the parcel tends to move back to the reference level, the 
atmosphere is said to be stable at that level ; and if the parcel tends to 
move away from the reference level, the atmosphere is said to be unstable 
at that level. This method of characterizing the vertical stability is 
called the parcel method. 

If the parcel were to mix with the surrounding air it would lose its 
identity, so we shall assume that the parcel is displaced without mixing. 
Moreover, although the displacement would actually disturb the environ- 
ment near the parcel, we shall further assume that the environment 
remains undisturbed. These assumptions are certainly very artificial, 
for the parcel must actually stir up and mix with the air through which it 
moves. It would be better to examine a small continuous displacement 
of a whole region. Then the stability criteria would depend upon the 
increase or decrease of the entire motion subsequent to the initial dis- 
placement. However, such an analysis requires advanced hydrody- 
namical technique and moreover gives the same stability criteria as the 
parcel method. The simpler parcel method will therefore be used here. 

It will be seen presently that the stability is determined by comparison 
of the densities of the parcel and of the surrounding air. Properties of 
the parcel will be denoted by primed symbols and properties of the envi- 
ronment at the same level will be denoted by unprimed symbols. 

Any change of a thermodynamic property of the parcel is an individ- 
ual variation, depending upon the process which the parcel undergoes 
and is designated by the differential symbol d (see section 4-10). Al- 
though the environment is assumed undisturbed by the motion of the 
parcel, the properties of the environment will generally vary from level 
to level. Accordingly, as the parcel moves to new levels its environ- 

125 



Section 5-01 126 

ment undergoes a variation which then is a spatial variation and is 
designated by the geometrical differential symbol d (see section 4-10). 

At the reference level < the parcel and its environment have exactly 
the same thermodynamic properties, for the parcel has yet to be sepa- 
rated from the environment. However, the properties of the displaced 
parcel at any other level will usually differ from the properties of the 
environment. 

Since the environment is in hydrostatic equilibrium, the resultant 
buoyant force at any level is zero. Thus taking the z component of 
equation 4-16(2), we have 

f>p 

O.-g-a-- 

In general the parcel will have a vertical acceleration i> z = z. By New- 
ton's second law of motion this acceleration is equal to the resultant 
buoyant force per unit mass. We shall now assume that throughout the 
entire motion the parcel will adjust its pressure to the pressure of the 
environment. Thus 

3 '*P 

Z= ~g- OL 

dz 
Elimination of dp/dz from the above equations gives 

/< \ a '~ a 

(1) z=g 

a 

This relation equates the acceleration to the resultant buoyant force 
acting on unit mass of the parcel. The buoyant force is expressed in 
terms of the specific volumes of the parcel and the environment. This 
force is upward if the parcel is lighter than the environment (a > a) 
and downward if the parcel is heavier than the environment (a < a). 

5-02. Stability criteria. The specific volume is not directly available 
from aerological data, so the buoyant force is more conveniently 
expressed by the virtual temperature. The equations of state 3-25(5) 
for the parcel and for the environment show that 5-01 (1) may be written 

T 1 *' _ T* 
(D 2 = g y; 

We shall investigate only infinitesimal displacements from the refer- 
ence level 00- The geometric height coordinate z will be measured from 
this level, so the dynamic height of both the parcel and the environment 
above the reference level is d<t> - d$ gz. Let the virtual temperature at 



127 Section 5-03 

<o be TQ . The virtual temperature variation in the environment is 

57"* 
(2) T* - 7 o* = 5r* = - 5* = -y*gz. 

6q> 

Here the virtual temperature lapse rate 7* in the environment is defined 
by the spatial derivative 

__* r *, 

7 50 * 

An individual virtual temperature lapse rale 7*' of the parcel may also be 
defined by the individual or process derivative 



d<t> 

The change of the virtual temperature of the parcel is then 

dT* f 

(3) - r*' - r * - dr*' - -d<t>=- y *'gz. 

d<t> 
Subtraction of (2) from (3) gives 

T*' - T* = g(y* - T*'). 
Equation (1) may then be written in the final form 

2 

(4) 2-^5 (7* - 7*')z- 

The reference level will be stable, indifferent, or unstable according to 
the following three conditions, read respectively from top to bottom : 

(5) 7* | 7*'. 

That is, the atmosphere is stable at the reference level if the acceleration 
and the displacement have opposite signs, so the parcel tends to move 
back to the reference level. The atmosphere is indifferent if the acceler- 
ation is zero. And it is unstable if the acceleration and the displace- 
ment have the same sign, so the parcel tends to move away from the 
reference level. 

If the virtual temperature sounding curve is available, we can easily 
determine whether any level of the sounding is stable, indifferent, or 
unstable, once the individual or process virtual temperature lapse rate 
is known. 

5-03. The individual lapse rate. The individual virtual temperature 
lapse rate in any process has been defined as 

, _dT*' 

y " d<t> ' 



Section 5-03 128 

where dT* f is the change of the virtual temperature of a parcel lifted 

through the dynamic height d<t>. Since it is assumed that the parcel 

travels through an undisturbed environment, the dynamic height is 

measured in the environment. The environment is in hydrostatic 

equilibrium, so 60 = d<t> = -adp. The individual virtual temperature 
lapse rate is then given by 



dr^\ 

dp/ 



a\ dp 

Here the derivative in parentheses is determined completely by the 
process, whereas the specific volume a. refers to the environment. How- 
ever, since we consider only an infinitesimal displacement from the 
reference level, where o! = a, the individual virtual temperature lapse 
rate for the parcel method depends only on the process. Thus 

7*' = 



If we replace T* f by T f , the above argument will also apply to the 
individual lapse rate of temperature, 7' = -dT'/dQ. As explained in 
2'23 the temperature changes caused by radiation, conduction, or mixing 
are slow compared with the changes caused by the vertical motion of the 
parcel. Consequently the process performed by the parcel displaced 
vertically from its reference level is nearly adiabatic. In the following 
we shall consider the process to be strictly adiabatic. The process is 
then either an unsaturated or a saturated adiabatic process. The 
individual temperature lapse rate 7' for an adiabatic process will be 
denoted by y u for an unsaturated parcel and by y s for a saturated parcel. 

Since the parcel does not mix with its environment, the mixing ratio 
of an unsaturated parcel has the constant value WQ throughout the dis- 
placement. Individual differentiation of 



3-25(4) 

with respect to dynamic height gives for an unsaturated adiabatic 
process 

(1) 7*'= (1 + 0.61w6)7. 

As explained in section 3-27 the process curves for an unsaturated adia- 
batic process nearly coincide with the dry adiabats. Therefore the lapse 
rate y u of temperature along an unsaturated adiabatic process curve is 
almost equal to the dry adiabatic lapse rate 7<*= l/c pd . See 4-20(3). 
The lapse rate of virtual temperature along the unsaturated adiabats is 



129 Section 5-04 

then given approximately by 

(2) 7*' y u 7d (unsaturated process), 

for the factor 1 + 0.61w in (1) is nearly equal to 1. 

When a saturated parcel is displaced upward it will perform a satura- 
tion-adiabatic process. Since condensation occurs, the mixing ratio w a 
of the parcel decreases. Differentiation of equation 3-25 (4) with respect 
to dynamic height then gives 

(3) 7*' - (1 + 0.61^)7, - 0.61 T* 



d<j> 

The lapse rate of temperature along a saturation -adiaba tic process curve 
is y 8 . The lapse rate 7*' may easily be visualized by constructing the 
virtual temperature curve of a saturated parcel whose image point 
follows a saturation adiabat. As the mixing ratio decreases the two 
curves become closer, so the virtual temperature of the parcel decreases 
more rapidly than the temperature of the parcel. This result is also 
evident from (3), for dw' 8 /d<t> is negative. The correction term 
-Q.61T'dw' s /d<t> is then positive. In the lower troposphere this correc- 
tion term may be as large as 10%. See equation 5 '11(4). For rough 
calculations, however, we may make the approximation 

(4) 7*' 7 (saturated process). 

An exact formula for y u and y 8 will be given later in section 5-10. 

In the following sections stability criteria will be presented in terms 
of the lapse rates of temperature yd along a dry adiabat and y 8 along a 
saturation adiabat, since these adiabats are usually drawn on meteoro- 
logical thermodynamic diagrams. If the exact form of the stability 
criteria is wanted, the individual virtual temperature lapse rate 7*' 
must be computed. 

5-04. The lapse rate in the environment. Although the lapse rates 
7*' and 7' for the process are always nearly equal, the lapse rates 7* 
and 7 in the environment may be quite different, for the vapor stratifi- 
cation in the atmosphere is arbitrary. Spatial differentiation of the 
relation 

3-25(4) T*= (l + 0.61w)r 

with respect to dynamic height gives 

(1) 7* - (1 + 0.61wh - 0.6ir-^- 

When 7* in the stability conditions 5-02(5) is expressed by (1), we 



Section 5-04 130 

find that the reference level is more stable than the temperature sound- 
ing indicates, if the mixing ratio increases with height; and it is less 
stable, if the mixing ratio decreases with height. In particular, the top 
of a cloud may be less stable than the temperature sounding indicates, 
especially if the cloud is below a dry inversion. For example, let us find 
what moisture distribution would make an isothermal atmospheric layer 
(7 = 0) have a virtual temperature lapse rate equal to the dry-adiabatic 
lapse rate (7*= l/c p d). Equation (1) then becomes 

to_ 1 

60 ~ ~~ Q.6lc pd f 

This variation is not too improbable, for the thickness &z of the layer 
having a constant variation of mixing ratio from the value w at the 
bottom to the value at the top is given by 

Q.6lc pd Tw 

Az = 

g 

For instance, the thickness of a layer saturated at the bottom and dry at 
the top, having a mean pressure of 90 cb and a mean temperature of 
10C, is 150 m . Such a layer might be found at the top of a stratus deck. 

5-05. Stability criteria for adiabatic processes. The stability cri- 
teria 5-02(5) for an infinitesimal adiabatic displacement can now be 
assembled finally. If the parcel is unsaturated the three stability con- 
ditions are, from 5-03(2), 

(1) 7* I 7d- 

And if the parcel is saturated the three stability conditions are, from 

5-03(4), 

(2) 7* I 7,. 

The graphical use of these criteria is illus- 
trated in fig. 5-05, which represents a small 
section of a thermodynamic diagram. The 
virtual temperature at the reference level is 
represented by the point PQ. Through PQ 
the dry and saturation adiabats aj and a* 
are drawn . Any virtual temperature sound- 
ing curve c* through P* may be classified with respect to these adi- 
abats. The five possible classifications, illustrated in fig. 5-05 by the 
curves c 88t c^, us ut tm arer 

(55) The sounding curve c 88 is said to be absolutely stable if above the 
reference level it is warmer than the saturation adiabat a*, that is, if 




131 Section 5-06 

7* < T ; for then the reference level is stable no matter whether the air is 
saturated or unsaturated. 

(is) The sounding curve c t * is said to be saturated indifferent if it 
coincides with the saturation adiabat a* , that is, if 7* = 7,; for then the 
reference level is indifferent if the air is saturated and stable if the air is 
unsaturated. 

(us) The sounding curve c^ is said to be conditionally unstable if it 
lies between the saturation adiabat a* and the dry adiabat aj, that is, if 
7d > 7* > 7 ; for then the reference level is unstable if the air is satu- 
rated and stable if the air is unsaturated. 

(ui) The sounding curve c*^ is said to be dry indifferent if it coincides 
with the dry adiabat 0%, that is, if 7* = 7^; for then the reference level is 
unstable if the air is saturated and indifferent if the air is unsaturated. 

(uu) The sounding curve c^ u is said to be absolutely unstable if above 
the reference level it is colder than the dry adiabat #, that is, if 7* > 7^; 
for then the reference level is unstable no matter whether the air is 
saturated or unsaturated. 

Notice that these definitions refer only to the slope of the sounding 
curve c* at the reference level and do not give the actual stability at that 
level, because the stability depends upon whether the air is saturated or 
unsaturated. 

5-06. Stable oscillation. If the reference level is stable, the accelera- 
tion of the parcel will counteract the displacement. Therefore the parcel 
will eventually stop and be driven back by the buoyant force to its 
equilibrium or reference level. However, its inertia will carry it past 
this level. Consequently the parcel will oscillate about the equilibrium 
level. 

This can be shown analytically, for if the parcel is stable, a positive 
number v 2 can be defined as follows: 



Equation 5-02(4) may then be written 

2+ A=0. 

The solution of this second-order differential equation is well known. It 
is the equation for a simple harmonic oscillator. Thus 

z *= A sin vt. 

That is, the parcel oscillates about its equilibrium level (z - 0) with the 
amplitude A and the circular frequency v. The period r or time required 



Section 5-06 132 

for one complete oscillation is 



Evidently the more stable the reference level is, the smaller the period 
will be. As an example let us find the period of oscillation in a dry iso- 
thermal layer. We then have 7* = 0, 7*' = l/c p d, so (1) becomes 

27T 



g 

If the temperature TO is 0C, the period is 335 sec. However, since the 
lapse rate is usually nearer the dry-adiabatic lapse rate, the period of 
oscillation is in general much longer. 

5-07. Finite displacement. So far we have characterized the stabil- 
ity of the reference level for an infinitesimal displacement. We shall 
now investigate the stability of an atmospheric layer for a finite displace- 
ment of a parcel from its reference level. As before, we shall assume 
that the parcel does not disturb the environment and does not mix with 
the environment. These two assumptions are physically contradictory. 
If the parcel is small enough not to disturb the environment, it will 
rapidly mix and lose its identity. On the other hand, if the parcel is 
large enough to retain its identity throughout a finite displacement, its 
motion will cause compensating currents in the environment. Never- 
theless, it is useful to examine this fictitious case as a first approximation 
to the theory of convection. Later in section 5-09 we shall allow for 
compensating motion in the environment. 

The analysis of section 5-01 holds for any level <t> reached by the dis- 
placed parcel. Therefore the equation 

dv z a a 

5-01(1) z = g 

at a 

is valid for the parcel even after it has been displaced through a finite 
vertical distance. When the acceleration on the left-hand side of this 
equation is multiplied by the height increment dz, we find 

/^ j dv * j 

(i) *--,#. 

This is the change of the kinetic energy per unit mass of the parcel while 
it moves through the height dz. When the buoyant force on the right- 
hand side is multiplied by dz, we find 

a a a a f 

a ~~ a 



133 



Section 5-08 



This is the work done by the buoyant force on unit mass of the parcel 
while it moves through the height dz. As shown in fig. 5-07, dA is the 
area on an (,-/>) -diagram (or any other equivalent-area diagram such 
as the tephigram or the emagram) of the isobaric strip dp between the 
virtual temperature sounding curves c*', c* for . ., 

the process and for the environment respectively. / 
Since the expressions (1) and (2) are equal, we ^~ 
have by integration 






(3) 



v 2 zb - ) = - 




Thus the change of kinetic energy for any part of 
the finite displacement is equal to the area A be- 
tween the pressures p a and pb at the limits of the 
displacement, as shown in fig. 5*07. The area A is 
positive if the process curve c*' is warmer than the 
sounding curve c*, and negative if the process FIG. 5-07. Work per- 
curve is colder. Therefore when a parcel ascends formed on ascend ing par- 
through a colder environment its kinetic energy cel b v b "yant force, 

j i r . A j shown on tephigram. 

and, hence, its rate of ascent increase. And & 

when a parcel ascends through a 
warmer environment its kinetic 
energy and rate of ascent de- 
crease. 



5-08. Latent instability. We 

shall now examine the stability 
of a conditionally unstable layer 
with respect to a finite displace- 
ment. Fig. 5*08a shows such a 
layer schematically represented 
on a tephigram. Every level of 
the sounding curve c* from P 
to Q* is conditionally unstable. 

Consider the individual proc- 
ess curves c', c*' of an unsatu- 
rated parcel displaced upward 
from the re f er ence level P , PQ. 




Lifting 

condensation 
> N level 



Level of free 
convection 



FIG. 5-08<z. 



Reference level 



Latent instability, shown on 
tephigram. 



will follow the dry adiabat ad upward from P to the characteristic 
point P 9 , where the parcel becomes saturated (see section 3-28) . 



Section 5-08 134 

The level where this point is reafched is called the lifting condensation 
level. At the same time the virtual temperature image point of the 
parcel will follow the dry adiabat aj upward from P* to the virtual 
temperature characteristic point P*. Since the sounding curve c* is 
conditionally unstable, the parcel ascends through warmer environment, 
so its kinetic energy and rate of ascent decrease. If the parcel did not 
receive a sufficiently strong impulse at the reference level to reach the 
level P 8 , it would sink back toward the reference level. However, if the 
parcel is still ascending at the lifting condensation level, its image point 
will subsequently follow the saturation adiabat a s upward from P 8 . At 
the same time its virtual temperature image point will follow the satura- 
tion adiabat a s upward from P*. 

Notice that the curves c*, c*' intersect at the point P*. The area 
PjP*P* represents the decrease of the parcel's kinetic energy during the 
ascent. If the kinetic energy of the initial impulse is less than this area, 
the parcel will sink back toward the reference level. But if the kinetic 
energy of the initial impulse is greater than this area, the parcel spon- 
taneously will ascend beyond P* through a colder environment. Conse- 
quently the atmosphere is unstable with respect to the reference level if 
the initial impulse is strong enough for the parcel to reach the level P*. 
This level is called the level of free convection with respect to the reference 
level. And the atmosphere above the level of free convection is said to 
be latently unstable with respect to the reference level (Normand, 1938). 
Above the level of free convection the parcel will avScend through a 
colder environment and gain kinetic energy until it reaches the level R* 
where the curves c*, c* f again intersect. The parcel will arrive at this 
level with kinetic energy equal to the area enclosed by the curves c*, 
c*' plus the kinetic energy surplus at the level of free convection. At 
the expense of this energy it can penetrate into the stable region above 
-R* until all its kinetic energy is consumed. 

The level of free convection of a given reference level is determined by 
the point of intersection between the virtual temperature sounding curve 
and the saturation adiabat through the virtual temperature characteris- 
tic point. If the parcel is close to saturation at the reference level, the 
vertical distance to the lifting condensation level and also the distance 
to the level of free convection are small. So the kinetic energy of the 
initial impulse required to release the latent instability of the air at that 
level is also small. On the other hand, if the air at the reference level is 
far from saturation, the lifting condensation level is much higher and 
the corresponding saturation adiabat may not intersect the sounding 
curve, as indicated in fig. 508&. In this case no level of free convection 
and, hence, no latent instability exist with respect to the reference level. 



135 



Section 5-09 



The layers of latent instability, if any, are conveniently determined 
when both the virtual temperature sounding curve and the virtual 
temperature characteristic curve are plotted, as shown in fig. 5-08c. 
The virtual temperature characteristic curve c* connects the virtual 
temperature characteristic points P* corresponding to each point P* of 



Lifting _^ 
condensation P 8 
level 




Reference level s 



FIG. 5-086. 



FIG. 5-08c. 



c*. The layer of latent instability is determined by the point Q* on c* 
at which a saturation adiabat is tangent. This saturation adiabat inter- 
sects the virtual temperature characteristic curve at some point P*! 
corresponding to the reference level P\ . Latent instability exists for all 
levels below the level Pj, for the saturation adiabats corresponding to 
these levels intersect the sounding curve c*. However, latent instability 
does not exist for any level above the level P^, for the corresponding satu- 
ration adiabats do not intersect the sounding curve c*. 

5-09. The slice method. One shortcoming of the parcel method is 
that the effect of compensating motion of the environment is neglected. 
We shall now allow for this effect by considering an extended region of 
the atmosphere at a given reference level. The method for studying 
such a region was first developed by J. Bjerknes (1933) and is called the 
slice method. 

Let the region initially be completely saturated and also be homo- 
geneous in all thermodynamic properties at any level near the reference 
level 00- Within the region there may be several columns of ascending 
motion. In the remaining part of the region the air surrounding these 
columns will have a compensating downward motion, as indicated in 
fig. 5'09a. Let the total area with ascending motion be A ' and the aver- 
age speed of ascent be v f z . And let the total area with descending motion 
be A and the average speed of descent be v 8 . We shall further assume 



Section 5-09 



136 



that the mass ascending through the reference level in the time inter- 
val dt is equal to the mass descending through in the same time 
interval. 

These assumptions represent a simplified model of the atmosphere at a 
level of initial cumulus development. The ascending air will form 
clouds by saturation -adiabatic condensation . And the ascending motion 
in the cloud columns is compensated by descending motion in the sur- 



1 


r 




\ I I 





1 


t 


\v' 




\ i * 


1 


1 


A 


A' 


A 


A' 


A 


FIG. 5-09a. 



rounding air. The atmosphere at the reference level 0o will be called 
stable if the cumulus growth is retarded, indifferent if the cumulus growth 
is neither retarded nor accelerated, and unstable if the cumulus growth 
is accelerated. 

In the time interval dt an atmospheric slice of area A and height 
dz = v z dt will be transported downward through the reference level. 
The volume of this slice is dV = Adz and its mass is 



CD 

Substituting here dz 
(23 



dM = pAv z dt = pAdz. 

d<t>/g and subsequently pd<t> = dp, we have 
pAd<(> Adp 



dM- 



g 



The mass of the ascending slice may be obtained by a similar argument. 
Thus dM' is given by equations similar to (1, 2) wherein symbols refer- 
ring specifically to the ascending slice are primed. 

Initially the thermodynamic properties are homogeneous throughout 
the reference level, so p' = p. Therefore, from (1, 2), the initial ratio 
dM'/dM is equal to any one of the following expressions : 

A A'dz A'd<t>' A 'dp' 



____ _ _ 

dM Av z Adz Ad(j> Adp 

Since the upward and downward transports of mass through <o are 
assumed equal, dM f = dM. Accordingly, we have from the above 
equations 

A f v z dz_ d<t> dp 
( ' ^ / = '^' r 



For simplicity we shall assume that there is no horizontal advection 



137 Section 5-09 

within the region, so local temperature changes must be caused entirely 
by vertical motion. As in the previous sections the individual tempera- 
ture changes may be considered adiabatic. Consequently the ascending 
air will be cooled saturation-adiabatically, whereas the descending air 
will be heated dry-adiabatically. The individual lapse rate in the 
columns of ascending air is then y 8 , and the individual lapse rate in the 
descending air surrounding the cloud columns is 7^. 

So far nothing has been implied as to the distribution of temperature 
near the reference level except that the lapse rate 7 is the same both in 
the cloud columns and in the surrounding air. Although the analysis 
can be carried through for any value of 7, the most interesting case 
occurs when the reference level is conditionally unstable, that is, when 
yd > 7 > 7- The following analysis will be confined to this case. 




dp 



\ r ' \ r ' 



. C- 

FIG. 5-096. 

As shown in fig. 5-096, the sinking air which started from the level 
<o + d$ with the initial temperature T = T - yd<t> descends dry-adia- 
batically and arrives at the reference level < with the temperature 

(4) Ti = T + y d d<t> = To + (TH - 7)d*- 

Since 7d > 7, T\> 7V Therefore at the reference level the region A 
becomes warmer owing to the subsidence. 

During the same time interval the ascending air which started from 
the level <fo - d^ with the initial temperature T 1 = TO + yd<t>' cools 
saturation-adiabatically and arrives at the reference level with the tem- 
perature 

(5) T( - T 9 - y*d<t>' - T + (7 - y s )d<t>'. 

Since 7 > 7, T( > T Q . Therefore at the reference level the region A f 
becomes warmer owing to release of latent heat. 



Section 5-09 138 

As in the parcel method, the stability is determined by the buoyancy of 
the cloud columns relative to the surrounding air. Since both regions 
are initially saturated and at the same pressure, the buoyancy depends 
only upon the temperatures T( and T\. The atmosphere at the -refer- 
ence level is stable, indifferent, or unstable according to the following 
three conditions, read respectively from top to bottom : 

(6) 21 1 2V 

From (3) we have d<t>~A f and d<j> r ~A. Therefore, when equations 
(4, 5 ) are substituted in (6) , the three stability conditions may be written 

(7) A(y - 7 5 ) ^ A' (yd 7)- 

A useful form of (7) is obtained by evaluating the lapse rate 7* for 
which the convection is indifferent. This critical lapse rate is the 
weighted average of the lapse rates 7^, y s . Thus 

(8) 7 < ' *~ 



Jt ~ A' + A 

The final form of the three stability conditions is simply expressed in 
terms of the critical lapse rate 7,- as follows: 

(9) 7 | 7.'. 

This form of the stability conditions for the slice method is similar to 
the form of the stability conditions for the parcel method. However, in 
the parcel method the critical lapse rate is yd if the air is unsaturated, and 
it is 7 S if the air is saturated. Equation (8) shows that when the air is 
conditionally unstable the critical lapse rate for the slice method lies 
between these two extremes; that is, yd > 7,- > 7 S . 

Evidently the slice method can be more generally applied than the 
parcel method. Indeed, the results of the parcel method may be 
obtained as special cases of the slice method. Let the area of the region 
A 1 + A be constant. Then, as A 1 0, each finite ascending column 
contracts into an infinitesimal filament ascending saturation-adiabati- 
cally through an undisturbed environment, and from (8) 7* y 8 . As 
A > 0, the finite descending region becomes an infinitesimal filament 
descending dry-adiabatically through an undisturbed environment, and 
from (8) 7i > yd- Therefore, since the lapse rate 7 of the region lies 
between 7* and 7^, the cumulus convection is accelerated (7 > 7,-) if the 
areal ratio A* /A is small enough. And the cumulus convection is 
retarded (7 < 7*) if the reciprocal ratio A/A' is small enough. 

So far we have assumed that the regions of ascending and descending 
motion are given. Actually the extent of these regions can be deter- 



139 Section 5-10 

mined only after the cumulus development starts. A perturbation with 
sufficiently narrow columns in an initially saturated reference level would 
be unstable. Moreover, the perturbation is more unstable, the narrower 
the columns. However, a very narrow ascending air column would 
rapidly be destroyed by turbulent mixing. Therefore the ascending 
columns must have a certain finite minimum cross section in order to 
maintain the upward convection. The column cross section actually 
selected by the perturbation must be small enough to be unstable and 
yet large enough not to be destroyed by the consequent convection. 

As shown in fig. 5-096 the instability is insured by T( > T\. The 
difference T{ T\ required to maintain unstable convection may be 
quite small, whereas the difference T\ TO may be comparatively large. 
The effect of such convection would be to heat the reference level and all 
subsequent levels reached by the convective columns. This heating is 
supplied by release of latent heat of condensation in the ascending air and 
by subsidence in the descending air. Since all levels reached by the 
convection are heated, heat is transported upward by the convective process. 
This process is the most important means of transporting heat upward 
in the troposphere. 

At the highest levels reached by the convection the heat is distributed 
horizontally from the tops of the ascending columns. The heat which is 
transported downward to the levels below by subsidence is thereby con- 
tinually replenished by the latent heat released at the top of the convec- 
tive region. Accordingly, the heating of the entire region affected by the 
convection is ultimately supplied by the latent heat of condensation 
released by the upward convection. 

5-10. Formulas for y u and y 8 . In section 5-03 certain unsaturated 
and saturated process lapse rates y u and y s were defined. For the pur- 
pose of estimating stability y u can be closely enough approximated by 
y<i, and y s can be well enough estimated from a diagram. For other 
purposes, however, it is important to have exact formulas for these 
lapse rates. These will now be derived. 

The adiabatic process for unsaturated air is given by 

2-24(3) = c p dT - adp, 

where c p = c p d(l + 0.90w) is the specific heat of moist air at constant 
pressure. As in 5-03, the individual change of dynamic height is given 
by the hydrostatic equation: d<t> = -adp. Hence we have 

dT 1 
(1) 7u =- =-. 

(t(f> C p 



Section 5-10 140 

This formula shows that when w = 0, y u reduces to l/c p d, i.e., to yd* 
See 4-20(3). 

The value of y a at any point in the atmosphere depends on which of the 
saturation-adiabatic processes (see 3-31, 3-32, 3-34) the lifted air parcel 
is presumed to follow. We shall give the formula for the pseudo-adia- 
batic process. According to 3-31(4) this process is described by the 
equation 

(2) -Ldw s = (1 + w s ) (c p dT - adp). 

In order to evaluate y s , we must first express dw s in terms of dT and dp. 
We note from 3-20(4) that 

ee? 

(3) w.---^-. 

P - e s 

Differentiating (3) logarithmically, we find after collecting terms that 

... dw s p /de s dp\ 

iq.\ ---- = i -- _ i . 

w s P- e a \e s pi 
But from (3) we see that 

t) Wo 

(5) ~ JL - = 1 + -- = 1 + 1.6lw a . 

p - e s e 

Substituting (5) and Clapeyron's equation 3-11(4) into (4), we get 



This may be transformed further by noting that R = Rd(l + 0.61w): 
dw a 1 + 



.61wi,) RT 1 

" ' p P \ ' 



w s 1 + 0.61w 8 RdTl T 

When we simplify the right-hand side and use pa = RT, we get 
_ dw s (l + w 



T 

This is the second relation between dw s , dp, and dT referred to in sec- 
tion 3-33. By putting (6) into (2), canceling (1 + w a ), and collecting 
terms, we get 

f Lw a c/Xl + 



L RdT T 

We substitute d<t> - -adp, and introduce two abbreviations: 



,. Lw 8 eL(l -f- 0.61w 8 ) eL 

( 7 ) M-^-^;, "-- 



c p T c pd (l 



141 Section 5-11 

We then get 



Now 7a=-d7Vd0; and l/y u = c p , by (1). Hence -c p (dT/d<l>) 
7s/7u> This ratio is denoted by w: 

(9) n-^. 

7u 

We then get from (8) and (9) : 

1 + M 

(10) W = - 7a = "7*. 

1 + W 

This formula (10) is the desired formula expressing 7* in terms of the 
known meteorological variables of (1) and (7). 

From (10) we may compute n and hence 7., at any point on a diagram. 
It is seen that < n < 1. A typical value of n for the lower tropo- 
sphere is that determined for the state T = 275K, p = 90 cb, where 
w s = 5 X 10""'*. From (7) we obtain p. = 0.159, v = 5.62, whence from 
(10) n = 0.61. This corresponds to 7 S = 0.00061. This is a tempera- 
ture drop of approximately 6C per kilometer rise, which is accordingly 
used as a value of the saturation-adiabatic lapse rate for many rough 
calculations. The minimum value of y s found in the atmosphere may be 
estimated as that at T = 309K, p = 100 cb, w 8 = 40 X 10~ 3 , for which we 
compute 7 S = 0.000325. 

Formula (10) is valid for either the rain stage or the snow stage of the 
pseudo-adiabatic process (see 3-30). The only difference is that in the 
snow stage the latent heat L of evaporation in (7) should be replaced by 
the latent heat Li v of sublimation. 

5-11. Rate of precipitation. A useful application of the saturation- 
adiabatic lapse rate is the evaluation of the amount of precipitation which 
falls from an ascending saturated layer. The assumption is made that 
the process is pseudo-adieibatic (see 3-30), so that all the condensed 
water falls out as precipitation. The method, given below, of finding 
the rate of precipitation was developed by Fulks (1935). 

Consider a saturated sample of air containing one ton of dry air. It 
will contain w 8 tons of water vapor and thus its total mass is (1 + w 8 ) 
tons. If this mass is lifted one meter, dw s /dz tons of vapor will be 
condensed. The amount of precipitation per ton of saturated air when 
lifted one meter is therefore 

(1) 1 dw . 

1 + w 8 dz ' 



Section 5-11 142 

Consider next a vertical column of saturated air having the cross- 
sectional area of one square meter and the dynamic height of one dy- 
namic decimeter. The volume of this column is l/g cubic meters, and its 
mass is p/g = I/ (ag) tons. Let P' be the number of tons of precipitation 
from this column when it is lifted one meter. From (1) we then have: 

(2) P'- * dw 8 ^_ 1 dw a ^ 

(1 + w 8 )ag dz (1 + w 8 )ot d<t> ' 

To evaluate dw 8 /d<t> we divide 5-10(2) by d<t>: 
(3, *- + 



By using d<t> = adp, and n = c p (dT/d(t>) from 540(9), we can solve 

(3) for -dw a /d<t>: 

/*x dw * ,< \ C 1 "" n ) 

(4) (!-*- w 8 ) 

From (2) and (4) we get the final formula for P 7 : 

(s) P '"^r' 

where n is given explicitly by 540(10). As a dimensional check of (5), 
we note that P' is defined to be a mass, per dynamic unit column 
expressed in (meters) 2 X dynamic decimeters, per meter lift. Hence 
we should have 



The right side of (5) has these dimensions, since n is a pure number. 

We now define P to be the rate of precipitation in millimeters per hour 
from a saturated layer 100 dyn m thick ascending at the speed of one 
meter per second. Obviously P is independent of the area of the surface 
on which the precipitation falls, so we shall consider one square meter 
on the ground. Above this area are 1000 columns like that of the previ- 
ous discussion. In one second they will rise one meter, and each will 
drop P* tons of precipitation. In one hour they would at the same rate 
collectively provide 3.6 X 10 6 P' tons of precipitation. One ton of water 
in a unit column will have a height of one meter, or 1000 mm. Conse- 
quently, in one hour this 100-dyn m layer will give 3.6 X 10 9 P' mm of 
precipitation. We thus have from (5) : 

(6) P - 3.6 X 10 9 ^^ mm hr" 1 . 

oL 



143 



Section 5- 12 



Formula (6) is the practical form of the precipitation equation. In 
general, if the saturated layer were AH dynamic meters thick, rising at 
the speed of v z meters per second, then the rate of precipitation would be 

<" 



For the snow stage L in (6) should be replaced by ,,-. 
30 




10 20 30 

FIG. 5*12. Precipitation lines on emagram. 



-30-20-10 
T 



5*12. Precipitation lines on the emagram. Since both n and a in 
541 (6) depend only on p and T, it is possible to draw lines of constant P 
on any thermodynamic diagram. They are shown in fig. 5-12 on an 
emagram, and the value of P is given for each line. There are two sepa- 
rate families of lines. Those marked with R are for the rain stage. 
Those marked 5 are for the snow stage. Both sets of lines are due to 
Fulks. 

We give a simple example of the use of the precipitation lines. Let us 



Section 5-12 144 

suppose that a saturated isothermal layer extends between the pressure 
levels 900 mb and 600 mb. The temperature of the layer is 10C, and it 
is known to have the vertical velocity 0.3ms" 1 . It is desired to estimate 
the resulting rainfall in millimeters per hour. 

We break the layer up into the three sublayers 900-800 mb, 800-700 
mb, and 700-600 mb. For each layer we perform the following calcu- 
lations: (i) We determine its thickness A/7 in dynamic meters using any 
convenient method; (ii) we determine the value of P at the midpoint of 
the sublayer from the precipitation lines; (iii) we determine the precipi- 
tation from that sublayer using formula 5-11(7). The resulting figures 
in (iii) are then added, to give the total rainfall in millimeters per hour. 
The calculation is summarized in the following table. 

SUBLAYER * , AH 

- 



, 
(m s- 1 ) (dyn m) 100 

900-800 mb 0.3 960 0.72 2.1 

800-700 mb 0.3 1090 . 67 2.2 

700-600 mb 0.3 1250 0.60 2.3 

Total rainfall in mm hr~~ l = 6.6 

In practice the determination of v z is quite difficult, and it limits the 
accuracy of the whole procedure. There is consequently little need of 
much accuracy in the determination of either A77 or P. 



CHAPTER SIX 
THE EQUATION OF MOTION 

6-01. Kinematics. In chapter 4 we examined the atmosphere in a 
state of relative or hydrostatic equilibrium. That is, we considered the 
atmosphere when it is rotating with the earth. We know from experi- 
ence that motion in the free atmosphere is much more complicated. In 
order to investigate this more complicated motion we must advert to 
kinematic and dynamic concepts. 

The description of motion is called kinematics. Consider a moving 
point. Let its position P be described from a reference frame which 
we may think of as a Cartesian coordinate system centered at some 
point O. P is then specified by a coordinate triple (x,y,z). 

However, another and equivalent way of representing P is to draw 
the directed vector length from to P. This vector is called the position 
vector of P and is symbolized by r. The x,y,z components of r are merely 
the distances x,y,z. Hence 

(1) r= xi + y) + zk. 

Here i,j,k may represent any set of rectangular unit vectors. In later 
applications we shall let i,j,k represent the standard system defined in 
4-03. 

6-02. Velocity. As the point moves along the path c shown in 
fig. 6-02, it occupies different positions at different times. At the time / 
let its position be P and its position vector be r = r(/) At the later time 
/i = t + A/ let its position be PI and its posi- 
tion vector be TI = r(/ + A) Corresponding to 
the time difference A/ = t\ /, the position vec- 
tor difference may be denoted by Ar = TI - r. 
Since the vector difference ri - r is the vector 
from the terminus of r to the terminus of TI, Ar 
is the vector from P to PI. 

The quotient Ar/A2 is also a vector. It has 

4-U J- *.* J A A 4-U 

the same direction and sense as Ar. As the 
time increment becomes smaller, P will approach PI, and the quotient 
will approach a limiting value. This limiting value is defined to be the 

145 




Section 6-02 146 

velocity at P, and will be denoted by v. We then have 

(1) v-lim- 1 - = lim . 

* 



This vector limiting process is quite analogous to the familiar scalar 
process of differentiation. Hence the usual symbols for derivatives may 
be extended to vectors.. So we may write 

(2) V= f =f ' 

where the dot placed above a symbol is equivalent to the operation d/dt 
upon that symbol. We shall continue to use the differentiation symbol d 
as we have used it here : to denote variation with respect to a designated or 
individual point. 

The approach of PI to P along the path c not only defines the velocity, 
but also defines the tangent line to the path. Consequently the velocity 
must be along the tangent. We shall call the unit vector along the path 
in the direction of the motion the unit tangent and denote it by t. Let 5 
represent a linear coordinate along the path, so that ds = \dt\. The 
unit tangent is then defined by 

dr dr 

Since the unit tangent is the unit vector along the velocity, we have 
v ut, where the numerical value of the velocity, called the speed, is 

(3) ,-W-t*-*. 
^ } ' ! \dt dt 

The rectangular components of the velocity may be obtained by sub- 
stituting the rectangular components of r\ and r into (1) and finding the 
limit. Since i, j, k are vectors fixed in the coordinate frame, the rate of 
change of the position vector may be expressed in terms of the rates of 
change of its components as follows : 

r = xi + yj + zk. 

Therefore the rectangular components of the vector derivative of a posi- 
tion vector are the scalar derivatives of the rectangular components of 
the position vector. 

6-03. Differentiation of a vector. Any vector a which is a continu- 
ous single- valued function of the scalar variable u may be represented as 
a position vector. Let the vector a be drawn from a given origin 0. 



147 Section 6-03 

As the variable u increases, the terminus of a traces a continuous curve, 
called the terminal curve, similar to the path traced by the terminus of the 
position vector. 
The vector derivative of a = a(w) is defined as 

/i\ da i- a i~a ,. Aa 

(1) = hm - = lim 9 

du Ul ^u HI- u Au __>o AM 

where Ui = u+Au and ai = a(u+&u). Evidently by definition 
da/du is tangent to the terminal curve. Thus the derivative of a vector 
is easily visualized when the vector is represented as a position vector 
and the terminal curve is drawn. 

By expanding ai and a into rectangular components we get for the 
rectangular components of the derivative: 



(2) . 
du du du du 

Therefore the rectangular components of the vector derivative of a are 
the scalar derivatives of the rectangular components of a. 

The rules for differentiating functions, sums, and products involving 
vectors correspond to the rules for differentiating functions, sums, and 
products of scalars. That is, 

... da dadw 

(3) 7~ = T" T"' w= * w ( w ) 

du dw du 

... d , . . da db ^ 

(4) ;r( a + b >-;r + ;r f 

du du du 

/e\ ^ / \ ^ e ^ a 

(5) T^-T^+tT 9 

du du du 

t*\ & f ^ &* t_ d** 

(6) (a-b)- -b+.a. 

du du du 

These rules are easily verified either by writing the derivatives as limits, 
according to (1), or by resolving the vectors into rectangular components 
and applying (2). 

The rule (6) can be used to show that the derivative of a unit vector is 
normal to that vector. For differentiation of aa = 1 gives 

da da da 

a + a- - 2a - 0. 
du du du 

This result could be anticipated geometrically, for the terminal curve of a 
unit vector is constrained to lie on the surface of a unit sphere. Since 



Section 6-03 148 



da/du is tangent to this terminal curve and consequently tangent to the 
sphere, it must be normal to a. 

All the definitions and results of this section also apply to partial 
derivatives symbolized by 




6-04. Acceleration. The position of a moving point has been 
described by the position vector r = r(/) issuing from an origin O. As 
time passes, the terminal point of the position vector moves along the 
path c with the velocity v = v(/). 

A clear picture of the variation of the velocity 
is obtained by transferring the origin of the 
vector v from the path point P to the origin 0, 
as shown in fig. 6-04. The terminal point P f of 
the transferred velocity vector is called the image 
point. As time passes, this image point traces a 
terminal curve c' called the hodograph of the 
velocity. To each instant / there corresponds a 
definite point P on the path c and a definite 
image point P f on the hodograph c '. P repre- 
sents the position of the moving point and P f represents its velocity. 

The velocity of a point is the individual rate of change t of the posi- 
tion vector. The individual rate of change of the velocity v = dv/dt 
will be called the acceleration of the point. The acceleration is readily 
obtained from the hodograph. For, as shown in the figure, the point P 
moves with the velocity f along the path c traced by r. And the image 
point P' moves with the velocity v along the hodograph c' traced by v. 
Accordingly, the acceleration of a moving point along the path is equal to the 
velocity of the image point along the hodograph. 

From 6'03(2) the rectangular components of the acceleration are 
given by 



(1) 



In the next section we shall derive another and more useful expression 
for the acceleration. 

6-05. Curvature. When the relation v = vi is differentiated we find 

(1) v= vt+ vi. 

Since the derivative of a unit vector is normal to that vector, t and t are 
perpendicular. Consequently the acceleration has here been split into 
components tangential and normal to the path. We shall call the unit 
vector along the derivative of the unit tangent the unit normal and 



149 



Section 6-05 



denote it by N. That is, 

All the normals to the unit tangent lie in the plane perpendicular to the 
path. To find the direction of the unit normal at a point P on the path, 
we must examine the motion along the path in the neighborhood of P. 

Suppose that the unit tangent t is 
drawn at P as in fig. 6'05a. Let us 
select another point PI near P and draw 
the unit tangent ti at that point. The 
vectors t and ti are of unit length. 
Therefore, as PI -> P, the length of the 
vector At = ti t becomes numerically 
equal to the angle A^ between t and ti- 
Thus 



(3) 



]*J 




1. 



FIG. 6-05a. 



The rate of change of the unit tangent may be expressed by means of 
(2), (3), and 6-02 (3) as 

where K = d\f//ds is called the curvature of the path c at P. Curvature is 
a geometric rather than a kinematic concept: it is measured by the 
angular turn A^ of the tangent through the arc length As. That is, the 
curvature of a curve at a point P on the curve is defined as 



(5) 



K 



lim - -f 
P.-+P As as 



Evidently this formal definition corresponds to the intuitive notion of 
curvature. For instance, as shown in fig. 6OS6, the curvature of a circle 
of radius R is 



ds 



R 



This result is important: The curvature of a circle is the reciprocal of the 
radius of that circle. A straight line may be regarded as a circle of infi- 
nite radius having zero curvature. 
The path in the neighborhood of P may always be replaced by a circle 



Section 6-05 



150 



having the same curvature as the path. Before constructing this circle 
let us first consider the construction of the tangent line at P. Choose a 
point PI on the path c near P. The two points P and PI determine a 
straight line. As PI - P this line approaches a limiting line tangent to 

the path at P. 

Consider now two distinct points, 
PI and P 2 , on the path c near P. 
The three points P, P it P 2 deter- 
mine a circle. As P 2 > PI -> P this 
circle approaches a limiting circle 
called the circle of curvature to the 
path at P. The plane of this circle 
is called the osculating plane. 

Suppose that the path contains the 
three points in the order P, P 1? P 2 . 
The unit tangent ti is defined by the 
approach of P 2 to P\ , and the unit 
tangent t by the approach of PI to 
P. Consequently, as P 2 > PI > P, 
ti and t become coplanar with the osculating plane. Since both unit 
tangents are along the path, their difference is directed toward the 
center of the circle of curvature. This center will be called more 
briefly the center of curvature. 

As P 2 -> PI - P the unit tangents ti and t define the curvature of the 
path and also the curvature of the circle of curvature. Hence both 
curvatures are the same. Accordingly 




FIG. 6-056. Curvature of a circle. 



(6) 






where K is the curvature of the path, and R, called the radius of curva- 
ture, is the radius of the circle of curvature. 

When the expression (4) for the rate of change of the unit tangent is 
substituted in (1), we find 



The tangential component of the acceleration is called the tangential 
acceleration. It is numerically equal to the rate of change of the speed. 
Accordingly a moving point speeds up or slows down only in the direction of 
motion. Notice that v is not the magnitude of the acceleration. 

The normal component of the acceleration is called the centripetal 
acceleration, for it is directed toward the center of curvature. The 
vector K of magnitude K and directed toward the center of curvature will 



151 Section 6-06 

be called the vector curvature. Thus, K = KN. Accordingly the cen- 
tripetal acceleration is v 2 K. 

The centripetal acceleration may be expressed in another form by 
extending the notion of angular speed from the rate of angular rotation 
along a fixed circle to the rate of angular rotation along the osculating 
circle. This generalized or instantaneous angular speed, denoted by 
co, is defined as 

, W 

(7) - 5r - 

As a consequence of this definition 

'ds dsd\l/ o) 



The vector R of magnitude R and directed away from the center of 
curvature will be called the vector radius of curvature. Thus R - KN. 
So the centripetal acceleration is -to 2 R. 

When the centripetal acceleration is expressed by v 2 K or 2 R, the 
acceleration becomes 



(9) v 

(10) v-t5t-oj 2 R, 

respectively. 

The unit tangent and the unit normal constitute an orthogonal system 
in which the expressions for velocity and acceleration are particularly 
simple. Any system so intimately connected with the motion will be 
called a natural system. And metric coordinates along the unit vectors 
of a natural system will be called natural coordinates. We shall make 
extensive use of such coordinates in the following. 

6-06. Reference frames. Let us denote by F the reference frame 
from which the motion of a point is described. The operations by which 
the position vector, the velocity, and the acceleration of the point have 
been obtained are defined only with respect to F. Suppose that the 
motion of the same point throughout the same interval of time is de- 
scribed from another reference frame F a . With respect to F a a position 
vector, velocity, and acceleration may also be defined. In general the 
position vector and its derivatives observed from F a will differ from the 
corresponding vectors observed from F. For each frame may move and 
change relative orientation in any arbitrary manner. 

Suppose that the frame F is rigidly attached to the earth. Such a 
frame is called a relative frame. Quantities referred to the relative frame 
will be called relative and their symbols will carry no subscript. 



Section 6-06 152 

Let the frame F a be attached to some point on the axis of the earth and 
be oriented so that the stars appear fixed. In this frame the axis of the 
earth also appears fixed. In dynamic meteorology such a frame is called 
an absolute frame (see 6*07). Quantities referred to the absolute frame 
will be called absolute and their symbols will be distinguished by the 
subscript a. 

In practice the motion of the atmosphere is always referred to a rela- 
tive frame. But, as we shall see in the following chapter, the dynamical 
description of the motion is referred to an absolute frame. 

6-07. Dynamics. The motion of material points or particles is not 
arbitrary but depends in a definite way upon the presence of other matter. 
The formulation and consequences of this dependence constitute the 
science of dynamics. We shall not define or develop the fundamental 
dynamical concepts; we shall suppose that these notions arc understood. 
Indeed, they have already been introduced and used in the preceding 
chapters. 

If mass and force are understood, then dynamics can be founded upon 
Newton's second law of motion, the fundamental dynamical law. This 
law may be formulated as follows : The resultant force acting upon a parti- 
cle is proportional to the mass of the particle, and is proportional to the 
acceleration of the particle as observed from an absolute frame. When units 
are properly chosen, as in the mts system, the acceleration and the 
resultant force per unit mass are equal. 

Newton's second law should properly refer the motion to a frame 
located at the center of gravity of the solar system and fixed with respect 
to the stars. Although this refinement is required for astronomical cal- 
culations, it is unnecessary for the dynamics of the atmosphere. We 
shall consider here that a frame located at a point on the axis of the earth 
and fixed with respect to the stars is an absolute frame, valid for the state- 
ment of Newton's second law. 

In the atmosphere the acting forces are the pressure force, the force of 
gravity, and the frictional force. Here we shall suppose that the fric- 
tional force is absent. Later in chapter 9 the effect of friction will be 
considered. The pressure force arises from the interaction of the air 
elements and is independent of the reference frame from which it is 
observed. However, as we have seen in 4-09, the force of gravity is 
measured upon the rotating earth and depends upon the rotation. 

6-08. The force of gravitation. An observer fixed in absolute space, 
not rotating with the earth, would measure the force of gravitation rather 
than the force of gravity. The force of gravitation arises from the 
attraction between mass points and is given by Newton's law of universal 



153 Section 6-08 

gravitation. The force of gravitation acting upon a unit mass at a dis- 
tance r from the center of the earth is directed toward the center, and its 
magnitude g a is given by 

, GM 

(1) fo--^-- 

Here G is the universal gravitational constant, and M is the mass of the 
earth. G has the dimensions [M"" 1 L 3 T W " 2 ], so it has the same value in 
both mts and cgs units. This value is 6.658 X 10~~ 8 . The mass of the 
earth has been found by astronomical measurements to be 5.988 X 10 21 
tons. 

Formula (1) holds only if the mass of the earth is symmetrically dis- 
tributed about the center of the earth in homogeneous concentric spheri- 
cal shells. Actually the earth is nearly an oblate spheroid of revolution 
slightly depressed at the poles. Moreover, the mass of the earth is asym- 
metrically distributed. Hence (1) cannot hold exactly. But it does 
give a good approximation to the actual force of gravitation. 

Since the earth is a spheroid, the polar and equatorial radii are differ- 
ent. The polar radius is a f > = 6357 km, and the equatorial radius is 
a E = 6378 km. A sphere with the radius a = 6371 km has very nearly 
the same volume and area as the earth. The value a will be used for the 
mean radius of the earth. Introducing this value into (1), we find the 
mean value of the force of gravitation at the surface of the earth to be 

/~* ii/r 

(2) g -- r - 9.822 m s~ 2 . 

a 

The force of gravitation is directed toward the center of the earth. 
If it has a potential, the equipotential surfaces must then be spherical. 
The normal distance between two infinitesimally separated spherical 
surfaces is dr. Consequently, if a gravitational potential <j> a exists, it 
must satisfy the relation 



When (1) is substituted for g aj the existence of the gravitational potential 
0o is proved, for 



Thus, by integration, 



Section 6-08 154 

If the constant of integration C is evaluated so that <f> a = at sea level at 
the poles, then 



(3) * a -GAf(--- 

\a P r t 

The gradient of the gravitational potential is the vector force of gravi- 
tation g a . Thus 

(4) go - -V0a- 

This force g a is the force of gravitational attraction of the earth acting 
upon a particle of unit mass. 

By the argument of 4-10 it can be shown that a gravitational poten- 
tial <f> a exists for any distribution of mass. Consequently the gravita- 
tional force g a is the gradient of this potential. That is, (4) holds for any 
distribution of the mass of the earth. 

609. The equation of absolute motion. The mathematical formula- 
tion of Newton's second law of motion is known as the equation of abso- 
lute motion. The forces observed from the absolute frame acting upon a 
particle of unit mass are the gravitational force g a and the pressure force. 

In section 4*15 the pressure force per unit volume was shown to be 
V/>. Consider the pressure force acting upon a parcel of mass 8M 
and of volume dV. This force is 5 Wp. The pressure force per unit 
mass, denoted by b, is then 

W 

(1) b= - V= -V. 

5M 

Newton's second law requires that the absolute acceleration v equal 
the resultant of the pressure force b and the gravitational force g . 
Thus! 

(2) v = b + go = -aVp - V</>. 

This is the equation of absolute motion referred to unit mass. The equa- 
tion of absolute motion referred to unit volume is obtained from (2) by 
multiplication with the density p. Thus 

(3) pv a = -Vp - pV0 - 

Although this equation is sometimes useful, hereafter the equation of 
absolute motion referred to unit mass will usually be intended when the 
equation of absolute motion is mentioned. 

The equation of absolute motion (2) may be expressed as an equilib- 
rium of forces by defining a force f a equal and opposite to the absolute 



155 Section 6- 10 

acceleration. Thus 

(4) fa = -V a . 

This force f a is called the inertial force of reaction. For it arises from the 
inertia of a particle moving relative to the absolute frame, and it com- 
pletely balances the resultant of all the acting forces. So the equation of 
absolute motion may be stated 

(5) 0=b + g a + f . 

That is, the resultant of all forces per unit mass, including the inertial 
force of reaction, is always zero. This formulation of Newton's second 
law is physically significant to an observer attached to the moving parti- 
cle, for he is unable to distinguish between real forces and the inertial 
force of reaction. Accordingly, whenever forces are measured upon a 
body accelerating relative to the absolute frame, inertial forces appear. 

The equation of absolute motion refers the acceleration to an absolute 
frame F a . Since these equations express so simply the dynamic con- 
ditions which control the motion, the dynamics of a process is often better 
understood when the motion is referred to the absolute frame. 

However, an observer fixed on the earth can express his observations 
more conveniently with respect to a relative frame F fixed to the earth. 
In order to use the equation of motion, observations relative to F must 
be referred to the absolute frame F a . That is, we must know how motion 
as observed from a relative frame would appear from the absolute frame. 
Evidently this depends upon the motion of the relative frame with respect 
to the absolute frame. 

6-10. The acceleration of a point of the earth. The earth rotates as 
a solid body from west to east with a constant angular speed which will 
be denoted by 12. 12 is then determined by the period of rotation of the 
earth with respect to the fixed stars. This period is called the sidereal 
day from the Latin word for star. Therefore, 12 is given by 

2ir rad 

ft = . 

1 sidereal day 

Since the earth moves around the sun, the sidereal day is different from 
the solar day or day, which is the period of rotation of the earth with 
respect to the sun. In one year, or approximately 365^ solar days, the 
earth has rotated 365^ times with respect to the sun. The earth has in 
one year also made one complete revolution in absolute space around the 
sun from west to east. Hence in one year the earth has made 366| 
revolutions with respect to the stars. Therefore 

1 year - 365 \ solar days - 366J sidereal days. 



Section 6- 10 156 

Consequently the angular speed of the earth is given by 



(1) _ ' = 7.292 x l<r 5 rad f 1 . 

365f solar days 

We may consider that the particles of the earth form a space of points 
called relative space. This space may be extended to include every point 
which appears at rest when observed from a point of the earth. Every 
point of relative space rotates with the constant absolute angular speed 12 
around the axis of the earth in a fixed circle of curvature centered on the 
axis. This circle is called a zonal circle and the plane of the circle, its 
osculating plane, is called a zonal plane. Since a point at rest in relative 
space moves with constant speed, its acceleration is purely centripetal 
and lies in the zonal plane, directed toward the axis of the earth. The 
absolute acceleration of a point fixed in relative space will be called the 
acceleration of a point of the earth and denoted by v e . Then by 6'05 (10) 

(2) v e =-fl 2 R. 

The equal and opposite inertial force of reaction f G is called the cen- 
trifugal force of a point of the earth. Thus, 

(3) f e =fl 2 R. 

So the equation of absolute motion for a particle at rest in relative space 
is, by 6-09(5), 

(4) 0=b+g a + f c . 

This is the equation of relative or hydrostatic equilibrium, expressed 
from the absolute frame. The same equation, expressed from the rela- 
tive frame, has already been obtained in 4'16(2) and is 

0=b + g. 

Evidently to an observer at rest in absolute space the pressure force is 
balanced by the force g -f f e - But to an observer at rest in relative 
space, the pressure force appears to be balanced by a single force, the 
force of gravity g. Accordingly we have 

(5) g=ga+f c . 

That is, as stated in the last section, a moving observer is unable to dis- 
tinguish between real and inertial forces. 

On a given zonal circle the magnitude f e = fl 2 J? of the centrifugal force 
is constant. The equiscalar surfaces of R are cylinders of revolution 
coaxial with the axis of the earth. So by 4-13 the ascendent of R is a 



157 Section 6- 10 

unit vector normal to those cylinders and directed away from the axis of 
the earth. Hence 



Since the angular speed of the earth is constant, the centrifugal force is 
the gradient of a centrifugal potential C . Thus 



where the centrifugal potential is given by 



The equipotential surfaces of e coincide with equiscalar surfaces of R; 
they are cylinders of revolution coaxial with the axis of the earth. 

Since both the gravitational force and the centrifugal force are poten- 
tial vectors, the force of gravity is also a potential vector. This was 
shown independently in chapter 4, where the potential of gravity was 
denoted by <. Equation (5) may then be rewritten in terms of poten- 
tial vectors as 



This equation shows that the potentials 0, < tt , and <j> e are related by 

(6) <t> = <t> a + <t> e . 

The potential <t> a in (6) may refer to any distribution of the mass of the 
earth. If, in particular, the earth is regarded as made up of homogene- 
ous spherical shells, then <f> a is given by 6-08(3), and (6) becomes 



(7) 

a P 

The geopotential surfaces of constant </> are found graphically by draw- 
ing spherical surfaces for <t> a and cylindrical surfaces for <j> e . Both sur- 
faces are surfaces of revolution which are completely defined by their 
intersections with any plane through the axis of the earth, called a 
meridional plane. Consequently the surfaces of constant < , e and < 
are completely represented by their traces on a meridional plane. The 
lines of constant gravitational potential are circles concentric with the 
center of the earth. And the lines of constant centrifugal potential are 
straight lines parallel to the axis of the earth. The diagonal curves of 
unit values of these two sets of lines form the meridional geopotential 
lines. 

The meridional traces of the potentials , </> c , and <t> are illustrated in 
fig. 6-10. Here the potential unit for which the traces are constructed is 
10 7 m 2 s~~ 2 . The shape of the geopotential surfaces has been exaggerated 



Section 6-10 



158 




-2.0 

FIG. 6-10. Geopotential levels. 



by drawing for a centrifugal potential which represents an angular speed 
ten times that of the earth. Notice that the geopotential surfaces are 
inflated in the direction away from the axis of the earth. In that direc- 
tion the centrifugal force becomes stronger and the gravitational force 

becomes weaker. Hence the dis- 
tance between consecutive cen- 
trifugal potential traces decreases, 
and the distance between consecu- 
tive gravitational potential traces 
increases. Accordingly the geo- 
potential surfaces far from the 
axis of the earth are nearly given 
by the cylindrical centrifugal po- 
tential surfaces, and the geopoten- 
tial surfaces near the axis of the 
earth are nearly given by the 
spherical gravitational potential 
surfaces. Although the geopoten- 
tial surfaces near the surface of the earth are actually inflated toward 
the equator, they are very nearly parallel, as explained in section 4-11. 
From (7) the surface < = coincides with the surface of the earth at 
the poles. This geopotential surface is found to be very nearly ellipsoi- 
dal with a difference of 11 km between the polar and equatorial radii. 
The values a P and a# obtained from geodetic measurement show that 
the difference should be 21 km (see section 6-08). This discrepancy is 
due to the assumption that the mass of the earth is distributed in homo- 
geneous concentric spherical shells. If, instead, $ a in (6) represents the 
potential of a homogeneous oblate spheroid, the geopotential surfaces 
will be more nearly correct. The remaining discrepancy, due mainly 
to irregularities of mass distribution in the crust of the earth, is small. 

6-11. Zonal flow. The equation of motion for a point rotating with 
the earth expresses the complete balance between the force of gravity 
and the pressure force. Accordingly, the isobaric surfaces must coincide 
with the geopotential surfaces. The ocean and the interior of the earth 
may be considered as viscous fluids rotating as a solid body with the 
absolute angular speed Q. So the isobaric surfaces in the earth and in 
the ocean are normal to the force of gravity. If the earth, the ocean, 
and the atmosphere rotate together as a solid body, the unit isobaric 
surfaces are then everywhere coincident with the geopotential surfaces. 
Since the thickness of an isobaric unit layer is proportional to the specific 
volume, the isobaric unit layers are very thin in the earth, thicker in the 
ocean, and much thicker in the atmosphere. 



159 Section 6-11 

Let us now examine an atmosphere rotating around the axis of the 
earth with the absolute angular speed co a . If co a is constant along each 
zonal circle and constant in time, the atmospheric flow is called zonal. 
In arbitrary zonal flow co a may vary from circle to circle. However, 
for simplicity we shall consider w a constant for all zonal circles. 

In zonal flow the acceleration is purely centripetal, so the inertial 
force of reaction is a centrifugal force, given by 

(1) fa==CO*R. 

And the equation of absolute motion is 

(2) 0=b 



When the atmosphere and the earth rotate together as one body, we 
have w = 12 and therefore f a = f e . Equation (2) then becomes identical 
with equation 640(4) for relative or hydrostatic equilibrium. 

Since the centripetal acceleration and the force of gravitation are 
meridional vectors, the pressure force and, consequently, the pressure 
gradient are also meridional vectors. Hence the isobaric surfaces must 
be normal to any meridional plane and are therefore surfaces of revolu- 
tion about the axis of the earth. The whole pressure field is then com- 
pletely determined by the meridional isobars, 
that is, by the lines in which the isobaric sur- 
faces intersect a meridional plane. 

The surfaces of constant gravitational poten- 
tial appear as concentric circles in the meridio- 
nal plane. Unless the absolute angular rotation 
is zero, the meridional isobars are depressed at 

the poles. This conclusion is illustrated by the 

_,. f , , ,. f ,. f FIG. 6-1 la. Meridional 

vector diagram of the equation of motion for isobar n zonal flow 

zonal flow in fig. 64 la. 

The centrifugal force o^R is the same for either sense of the absolute 
rotation about the axis of the earth. Consequently the isobaric sur- 
faces will be the same for either sense of rotation of the same magnitude. 
Although the two senses of the absolute rotation of the atmosphere are 
dynamically indistinguishable, they may be distinguished kinematically 
according to the sense of the earth's rotation. When the atmosphere 
rotates zonally with the same sense as the earth, o> a will be defined as 
positive. And when the atmosphere rotates with the opposite sense, 
co will be defined as negative. In order to conform with later sign con- 
ventions this definition will apply only to the northern hemisphere. 
However, the results for constant zonal flow may easily be extended to 




Section 6- 11 160 

the southern hemisphere, for both the field of rotation and the pressure 
field are symmetric about the equator. 

When co a is varied, the isobaric surfaces change orientation with respect 
to the surfaces of constant gravitational potential and of constant geo- 
potential. In the diagrams of fig. 6-116 four orientations of the isobaric 
surfaces, represented by the meridional isobars, are illustrated. Only 
one quadrant of the meridional plane is shown in the diagrams. The 
isobaric pattern is symmetric both about the equator and the axis of 
the earth. 

In diagram 6 the absolute angular speed is zero, and the atmosphere 
is in absolute equilibrium. The isobaric surfaces then coincide with the 
spheres of constant gravitational potential. In all the other diagrams 
the isobaric surfaces are depressed at the poles. 

In diagram b\ the centrifugal force of the atmosphere is less than that 
of the earth. Therefore the isobaric surfaces are less depressed at the 
poles than are the geopotential levels, and any constant level has a belt of 
low pressure at the equator. 

In diagram 63 the centrifugal force of the atmosphere is the same 
as that of the earth. The pressure force is equal, but opposite in 
direction, to the force of gravity. Therefore the isobaric surfaces 
coincide with the geopotential levels. 

In diagram 63 the centrifugal force is greater than that of the earth. 
So the isobaric surfaces are more depressed at the poles than are the geo- 
potential levels, and any constant level has a belt of high pressure at the 
equator. 

The absolute zonal rotation is observed from the earth as a relative 
zonal rotation with the relative angular speed co. Evidently the abso- 
lute angular speed co fl is the algebraic sum of the positive angular speed 12 
of the earth and the angular speed co relative to the earth. Thus 

(3) co = co H- fi. 

When co > the relative zonal wind blows from the west. When co * 0, 
the atmosphere is in relative or hydrostatic equilibrium. And when 
co < the relative zonal wind blows from the east. 

Relative zonal rotation in the same sense as the absolute rotation of 
the earth will be called cyclonic, and rotation in the opposite sense will be 
called anticyclonic. Zonal flow is then cyclonic when co > and anti- 
cyclonic when co < 0. Later the terms cyclonic and anticyclonic will be 
generalized. However, the conventions adopted here for zonal flow in 
the northern hemisphere will still hold. 

The diagrams 61, 6 2 , 63 of fig. 6-116 each represent both a positive and 
negative absolute rotation of the same strength. However, the pressure 



161 



Section 6- 11 






0<lo> a l<A 





C 2 





o>6>>-n 






C| 




FIGS, 6-116 and c. Pressure field in zonal flow. 



Section 6- 11 162 

field is not the same for positive and negative relative rotation of the 
same strength. In fig. 6-1 Ic the diagrams of fig. 6-116 are redrawn in 
order of decreasing co. Diagrams 6 lf b 2 , ^3 correspond respectively tc 
diagrams c\, c 2t 3 when the absolute rotation is positive, and to diagrams 
c'li c 2t c' 3 when the absolute rotation is negative. 

In diagram c 2 the relative angular speed is zero, and the atmosphere is 
in relative or hydrostatic equilibrium. The isobaric surfaces are then 
horizontal, so the pressure in any horizontal level is constant. 

Diagram c% represents the only case of cyclonic or positive relative 
rotation. Relative to the horizontal levels the isobaric surfaces slope 
downward from the equator to the poles. So any horizontal pressure 
field has low pressure at the poles and a belt of high pressure at the 
equator. As the cyclonic rotation increases, the meridional slope of the 
isobaric surfaces becomes steeper, and the strength of the horizontal 
pressure field increases. 

All the remaining diagrams in fig. 6-llc represent anticyclonic rota- 
tion of increasing strength taken in the order c\, CQ, c[, c^ 3. Evidently 
anticyclonic flow is more complex than cyclonic flow. Diagram c\ 
shows the anticyclonic rotation of moderate strength. Here the isobaric 
surfaces slope downward from the poles to the equator. So any hori- 
zontal pressure field has high pressure at the poles and a belt of low 
pressure at the equator. 

When the strength of the anticyclonic flow is increased, the slope of the 
isobars becomes steeper, until, as shown in diagram c 0t the critical value 
co = 12 is reached. Here the horizontal pressure field has a maximum 
strength with polar high pressure and equatorial low pressure, and the 
isobaric surfaces are spherical. 

Diagram c{ shows that the horizontal pressure field again becomes 
weaker with further increase of the anticyclonic rotation. At the second 
critical value co = 2S2 the isobars become horizontal, as shown in dia- 
gram c' 2 , so the pressure in any horizontal level is constant. 

Diagram c- 3 shows the anticyclonic flow increased beyond the value 
212. The isobaric surfaces then slope downward from the equator tc 
the poles as in cyclonic flow. Any horizontal pressure field has low 
pressure at the poles and a belt of high pressure at the equator. 

Zonal flow may be considered to have two centers: one at either pole, 
For cyclonic flow (see 3) the centers always have low pressure in any 
horizontal level. And for anticyclonic flow of strength less than -2fl 
(ci, CQ, c[) the centers have high pressure in any horizontal level. But 
for anticyclonic flow of strength greater than -212 (^3), the centers again 
develop low pressure in any horizontal level. 

Zonal anticyclonic flow of the types c{, c' 2 , 3 would never occur in the 



163 



Section 6- 12 



atmosphere. Although such anticyclonic flow is dynamically possible, 
no mechanism exists for generating absolute zonal rotation opposite to 
that of the earth. 

Later, when arbitrary horizontal flow is examined, it will be helpful to 
return to this section, for any horizontal flow will be shown locally to be 
similar to zonal flow. However the dynamics of zonal flow is simpler and 
should first be clearly understood. 

6'12. Angular velocity. In section 6-05 we defined the angular 
speed a for an arbitrary motion of a point. The plane of rotation is the 
osculating plane, and the axis of rotation is the line normal to the osculat- 
ing plane through the center of curvature. We shall now show how the 
rotation may be expressed vectorially. 

The rotation is completely specified by the numerical value of the 
angular speed, the orientation of the axis, and the sense of the rotation. 
Hence, a vector of magnitude o> directed along the axis of rotation accord- 
ing to the right-handed screw rule (section 4-03) expresses all the neces- 
sary information about the rotation. This vector is called the angular 
velocity and will be denoted by w. In particular, the angular velocity 
of the earth will be denoted by fl. Since the earth rotates from west to 
east, Q is directed from south to north parallel to the axis of the earth. 

The relation between the linear w 

speed v and the angular speed co is 
v = ojjR. The vector R of magnitude 
R is the radius vector directed from 
the center of curvature on the axis of 
rotation to the point whose motion is 
being considered. Let the radius vec- 
tor from any point on the axis of rota- 
tion to the moving point P be denoted 
by r. And let 6 (^ ?r) be the angle 
between the angular velocity o> and the 
radius vector r, as shown in fig. 612. 
radii R and r is then 

R = r sin 8. 

The velocity v is a vector normal to the plane of o> and r. Its magni- 
tude is 

v = cojR = (jor sin 0. 

And its sense is given by the right-handed rotation o> - r through the 
angle 6. Consequently the two vectors o> and r define the velocity com- 
pletely. Any vector similarly defined by two vectors is called the 
vector product of the two vectors. 




Axis of 
rotation 

FIG. 6-12. 

The relation between the scalar 



Section 6*13 



164 



6-13. The vector product. Let a and b be any two vectors, and let 
i TT) be the angle between them. The vector product v of these two 
vectors is indicated by a cross between them. Thus 

v= axb. 

The vector v is defined to be perpendicular to the plane through a and b 
and to have the magnitude 

(1) v - ab sin 0. 

The sense of v is given by a right-handed rotation a-b through the 
angle B. 

Let the vector b be projected into the plane perpendicular to a. This 
projection is a vector denoted by b# in fig. 6-13a. Its magnitude, given 
by bit b sin 0, is the altitude of the parallelogram having the base a 




FIG. 6-13a. Vector product. 

and the side b. So the numerical value of the vector product is equal to 
the area of the parallelogram, for v ab^. Hence the vector product may 
be considered as obtained by a positive rotation of b^ about a through a 
right angle and subsequent multiplication by a. 

When the order of the two vectors in the vector product is reversed, 
a new vector product is obtained whose magnitude still is given by (1). 
However the rotation b - a is opposite to the rotation a - b, so 

(2) bxa= -axb. 

Hence the vector product does not satisfy the commutative law but 
changes sign when the factors are permuted. 

If the two vectors a, b are parallel, the area of the parallelogram is 
zero, and the vector product is zero. And if the two vectors are per- 
pendicular, the vector product has the magnitude v * ab. Therefore the 



165 



Section 6-13 



(3) 



vector products of the rectangular unit vectors are: 

ixi=0, jxj = 0, kxk=0, 
ixj = k, jxk=i, kxi=j. 

Although the vector product is not commutative, it is distributive 
with respect to addition. Thus 

(4) ax (b-f c) = axb-f axe. 

The proof for this is demonstrated geometrically in fig. 6*136. All the 
vector products in (4) are in the plane normal to a. Suppose that this 
plane is the plane of the page and the vector a is directed toward the 
reader. The projection of a onto the page is the point labeled A . The 



vectors b and c project into the vectors b 
the triangular representation of the sum 
b + c projects as a triangle, we have 

(5) (b + C)AT - b N -f c^. 

The required vector products are obtained 
by positive rotation about a (counter- 
clockwise in the diagram) of the vectors 
in (5) through a right angle and multipli- 
cation by a. The vector equality (5) is 
not affected by the operations of rotation 



and c# respectively. Since 




FlG * 



and multiplication, so the distributive law (4) holds for the vector product. 
With the aid of this law we can obtain the rectangular components of 
the vector product. Let the two vectors be resolved into components 
and unit vectors. When the vector product is expanded by the distribu- 
tive law, and the vector products of the unit vectors i, j, k are reduced by 
(3), we find that the vector product a x b is given by 

(a y b z - a z by)i + (a z b x - a x b z }] + (a x b y - a y i x )k. 

This expansion may also be written as a determinant which is easier to 
remember. Thus 



(6) 



axb 



a-x 
b x 



The change in sign of the vector product with the reversal of the order 
of the factors appears here as a property of the determinant. For the 
interchange of two rows changes the sign of the determinant. 

The derivative of the vector product is obtained by differentiating (6) . 
Thus 



(7) 



d da , db 

(axb) = xb + ax 
du du du 



Section 6- 13 166 

This rule accords with the usual process of differentiating a product. 
Notice, however, that the order of the factors in the vector product must 
be preserved. 

6'14. The scalar triple product. Although only the properties of the vector 
product are required in this chapter, we shall later encounter the scalar prod- 
uct of a vector with a vector product. This scalar product is then of the type: 

a(b x c) = av = a v v. 

Since both factors in a vector product must be vectors, the parentheses enclos- 
ing the vector product in the above expression may be omitted. The scalar 
expression ab X c, involving three vectors, is called the scalar triple product. 

Geometrically this scalar represents the volume of the parallelepiped having 
the three vectors a, b, c as edges. This is easily seen from fig. 6-14. The 

vector product v = b X c is numerically equal 
to the area of the base parallelogram whose 
sides are b and c. The altitude of the par- 
allelepiped above this base of area v is the 

1- --y T - j I component a v of a normal to the base. So 
f ! / 1 the volume of the parallelepiped is a v v: the 

* ' product of its base and altitude. But a v v is 

also the value of the scalar triple product. 

The scalar triple product is positive when 
u a and v lie on the same side of the plane 

FIG. 6-14. Scalar triple product, through band c. If they lie on the oppo- 

site sides of this plane, the scalar triple prod- 

uct is negative. And if a lies in the plane the volume of the parallelepiped 
is zero, so the scalar triple product is also zero. 

The volume of the parallelepiped may be obtained by considering any two 
of the vector edges a, b, c as bases. By the right-handed screw rule all the 
rotations b- c, c~*a, a > b about a, b, c respectively have the same sense. 
Therefore, 

(1) ab x c = b-c x a = ca x b. 

Hence the scalar triple product does not change by cyclic permutation of its 
vectors. The non -commutative property of the vector product shows that 
the scalar triple product changes sign when the cyclic order of the vectors is 
altered. The scalar triple product is positive if the three vectors taken in 
cyclic order are relatively arranged as the axes of a right-handed coordinate 
system. 

Since the order of the factors in a scalar product may be reversed, we obtain 
from (1) 

(2) abxc = caxb = axbc. 




Hence the dot and cross can be interchanged without affecting the value of the scalar 
triple product. Geometrically this result is evident from the diagram. 



167 



Section 6- 15 



The scalar triple product may be expressed as a determinant of the rectangu- 
lar components of the three vectors a, b, c. From 6-13(6) we find 



(3) 



ab x c = a 



i ) k 

b x b v bg 

Cx Cy Cg 



CL X Q> 
b x b v 



Here the properties of the scalar triple product appear as the properties of the 
determinant. 

6-15. The velocity of a point of the earth. The concept of the vector 
product has been introduced for the purpose of expressing the velocity of 
rotation. We may now write 

(1) v= <oxr= coxR, 

where r is a radius vector from any point on the axis of rotation, and R 
is the radius vector from the center of curvature. 

In accordance with earlier conventions the absolute velocity of a point 
fixed in relative space will be called the velocity of a point of the earth and 
will be denoted by v e . This velocity is given by 

(2) v e = flxr = xR, 

where r is a position vector from any point on the axis of the earth, and 
R is the radius vector in the zonal plane. 

The velocity of a point of the earth is by definition the time derivative 
of its position vector. Let the position vector be drawn from the axis 



Axis of earth 
A 





FIG. 6-15a. 



FIG. 6-155. 



of the earth as in fig. 6-lSa. In the time interval dt the point moves from 
P to P e through the displacement d e r. The velocity of a point of the 
earth is then v 

d e r 



Section 6-15 



168 



The acceleration of a point of the earth is obtained by repeating the time 
differentiation on v e . Thus 



w a 

dt 
Introducing here the vector product (2) for v e , we find 

d e 
v e = (Q xr) = Oxv 6 . 

at 

It was shown earlier, in 6-10(2), that the centripetal acceleration of a 
point of -the earth is given by -12 2 R. It is readily seen from fig. 6-156 
that the vector product x v e has the same value. This vector product, 
perpendicular to fl and to v e , is in the zonal plane and is directed toward 
the axis of the earth. And its magnitude is $lv e Q?R. Consequently 
v e = -12 2 R. This expression for the centripetal acceleration of a point 
of the earth has here been independently verified. 

6-16. Absolute and relative velocity. In the last section we exam- 
ined the absolute motion of a point of the earth. The motion is that of a 
particle at rest relative to the earth. We shall now examine an arbitrary 




motion of a particle moving with respect to both an absolute and a rela- 
tive frame. 

Let the position of the particle be traced by a position vector r issuing 
from a point on the axis of the earth. Suppose the particle is located at 
the point labeled P in fig. 6-16. During the time interval dt let the point 
move to any arbitrary point P a - In the same time interval the point of 
the earth coinciding with the initial position of the particle will rotate 
around the axis of the earth from P to P e . We have already denoted the 
displacement from P to P e of a point of the earth by d e *- The displace- 
ment of the particle from P to P a , as observed from an absolute frame, 
will be denoted by d a f- And the displacement from P e to P a , as observed 



169 Section 6- 17 

from a relative frame, will be denoted by dr. The three displacements 
are shown in fig. 6-16. Evidently the absolute displacement is the 
vector sum of the displacement of the coinciding point of the earth and 
the relative displacement. Hence 

d a r = dr + d e r. 

Dividing this equation by the time interval dt required for the displace- 
ments, we find 

daT^dr dcT 

( } [dt "" dt dt ' 

The velocity of the coinciding point of the earth is 

d f r 
v e = = flxr. 

at 

So equation (1) may also be written: 

<*> =!+"< 

The rate of change with respect to the absolute frame of the position 
vector is by definition the absolute velocity, 



And the rate of change with respect to the relative frame of the position 
vector is the relative velocity, 

_dr 

~ dt 

Consequently equation (2) may be written 

(3) v = v+ v e = v + lixr. 



The absolute velocity of a particle is then equal to the vector sum of its 
relative velocity and the velocity of the coinciding point of the earth. 
The position vector r must issue from a point on the axis of the earth. 
For in the above derivation the origin of r must be fixed both in the 
absolute frame and in the relative frame. 

6*17. Absolute and relative acceleration. In section 6-04 it was 
shown that the acceleration of a point moving along the path may be 
considered as the velocity of its image point moving along the hodograph. 
When a particle moves in an arbitrary path, the hodograph of the abso- 
lute velocity may be obtained by drawing the absolute velocity vector 
issuing from an origin on the axis of the earth. This vector may be con- 



Section 6-17 170 

sidered as the position vector of the image point moving along the abso- 

lute hodograph. The relation between the absolute and relative veloci- 

ties of the image point may then be obtained in the same fashion as for 

the real particle when r in equation 6*16(2) is replaced by V . We then 

get 

/4N d a v a dv a 



Let the absolute velocity be expressed as the sum of the relative 
velocity and the velocity of the coinciding point of the earth. Then 
dVa/dt becomes 

d d dv 

- (v + v,) - - (v + Q x r) - + Q x v. 

at at at 

And Q x v becomes 

flx(v + v e ) = Qxv-fQxv e = Oxv + v c . 

The rate of change with respect to the absolute frame of the absolute 
velocity is by definition the absolute acceleration, 



And the rate of change with respect to the relative frame of the relative 
velocity is the relative acceleration, 



Therefore when the expressions derived above for dv a /dt and for 
fl x v are added we find 

(2) v - v + 20 x v + 1,. 

This equation relates the absolute and relative accelerations. It shows 
that the acceleration of a particle with respect to an absolute frame may 
be considered as the sum of three accelerations. The first term is the 
acceleration of the particle with respect to a relative frame. The last 
term is the centripetal acceleration of the coinciding point of the earth. 
The middle term is called the Coriolis acceleration. 

6-18. Absolute and relative zonal flow. A clear idea of the physical 
significance of the Coriolis acceleration, so called after its discoverer, is 
afforded by considering the special case of zonal flow. We have already 
discussed this simple type of flow (section 6-1 1). The relation between 
the absolute and the relative acceleration may be derived independently 



171 Section 6- 19 

for this motion without reference to the general theory. In accordance 
with the previous notation we shall let <o denote the absolute angular 
speed with sense and w the relative angular speed with sense. These 
two angular speeds are related by co = o> -f 12. 

The absolute centripetal acceleration of a particle is 

V a - -0>aR, 

and the relative centripetal acceleration is 

v = -co 2 R. 

If the absolute centripetal acceleration is expanded we find 
-o> 2 R - -(co + 12) 2 R = -co 2 R - 2coS2R - 12 2 R, 

or, by substitution of the absolute and relative centripetal accelerations, 

v a - v - 212R + v e . 

Comparison of this equation with 6*17(2) shows that the third term 
is the Coriolis acceleration; thus 

2flxv= -2col2R. 

This equation is readily verified with the aid of fig. 6*156, when in that 
diagram v replaces v e . 

6-19. The equation of relative motion. When flow in the atmos- 
phere is observed from a relative frame, the absolute acceleration is 
obtained from the relative motion by the relation 

6-17(2) v = v+ 2flx v+v e . 

The equation of absolute motion is 
6-09(2) v =b + go . 

When the absolute acceleration is eliminated between these two equa- 
tions, we find 

tf = b- 2Qxv4-g - V e . 

This equation has a clear meaning according to the principle of inertial 
forces : The acceleration relative to the earth is equal to the sum of all the 
forces, including the inertial forces arising from the absolute motion of 
the relative reference frame. The inertial force v e has already been 
called the centrifugal force of a point of the earth. Moreover, it has 
been shown in 6-10(5) that the sum of the last two terms in the above 
equation constitutes the force of gravity g. Hence, 

(1) *=b-2 



Section 6- 19 172 

The inertial force -20 xv is called the Coriolis force. This force is 
equal and opposite to the Coriolis acceleration. Since the Coriolis 
force acts normal to v, it has no component along the relative motion. 
Consequently it cannot contribute to the speeding up or slowing down of 
a particle relative to the earth. For this reason the Coriolis force is 
often called the deflecting force. The Coriolis force will be denoted 
more briefly by c, 

(2) c=-2Qxv. 
Equation (1) may then be written 

(3) v = b + c +"g. 

This equation is called the equation of relative motion, for it expresses 
Newton's second law of motion with respect to observations from a rela- 
tive frame. The equations of absolute and relative motion are com- 
pletely equivalent. Both describe the same motion, but from a different 
viewpoint. And both yield the same physical result for any given prob- 
lem. But when the problem admits of clear visualization in absolute 
motion, the equation of absolute motion is simpler to use, and the dy- 
namics is easier to understand. For this reason zonal flow is described 
from the absolute frame. In general, however, the equation of relative 
motion is used, for it is usually too difficult to visualize a given relative 
flow from the absolute frame. 



CHAPTER SEVEN 
HORIZONTAL FLOW 

7*01. Horizontal flow. Observations show that every large-scale 
current in the atmosphere is nearly horizontal. Appreciable vertical 
motion is usually confined to local convective currents, to boundary 
regions separating different air currents, or to mountainous regions where 
the air flow is influenced by topography. Regions of ascending air 
currents are often marked by the formation of clouds and precipitation 
due to lifting. These regions of vertical motion will be discussed later. 
We shall now examine flow which is strictly horizontal. 

The equations of motion derived in the last chapter are certainly valid 
for horizontal flow. However, these equations do not account for the 
frictional forces acting near the surface of the earth. Consequently we 
may apply the results of this chapter on horizontal flow to real currents 
only when the flow occurs in the free atmosphere above the friction layer 
and when the regions of strong vertical motion are excluded. 

The simplest type of horizontal flow is zonal. We shall find that many 
properties of zonal flow are also properties of arbitrary horizontal flow. 

7-02. Natural coordinates for horizontal flow. The plane tangent to 
the level surface is called the horizontal plane. Evidently at the point of 
tangency this plane is normal to the force of gravity and to the unit 
vector k directed toward the local zenith. Since the unit tangent is in 
the horizontal plane, t and k are orthogonal. 

In describing horizontal motion it will be convenient to introduce a 
unit vector orthogonal both to t and to k. This vector is a horizontal 
vector normal to the motion and is called the horizontal unit normal. 
The horizontal unit normal will be denoted by n, and the linear coordi- 
nate along it will be denoted by n. The three coordinates along the unit 
vectors t, n, k are then s, n, z respectively. The sense of the horizontal 
unit normal will be chosen so that t, n, k is a right-handed triple having 
the same relative arrangement as i, j, k. Accordingly the horizontal unit 
normal, viewed from the zenith, points to the left of the unit tangent. 

Notice that the horizontal unit normal n is in the horizontal plane and 
directed to the left of the motion; whereas the space unit normal N is in 
the osculating plane and directed toward the center of space curvature. 

The coordinate system described above constitutes a natural orthog- 



Section 7-02 174 

onal coordinate system. When the flow is toward the east, this system 
coincides with the standard orthogonal coordinate system described in 
section 4 -03. Both systems change orientation from point to point on 
any level surface. Hence they are valid only in the immediate neighbor- 
hood of the origin of the coordinates. For this reason they are called 
local coordinate systems. 

7'03. Standard and natural components. The equation of relative 
motion is a vector equation. This equation is equivalent to three scalar 
equations along three non-coplanar lines. We shall resolve the equation 
of relative motion along the three natural coordinates s, n, z and also 
along the three standard coordinates x t y, z. In both systems the vertical 
coordinate z is, of course, the same. 

Let a be an arbitrary vector. Let 1 be a unit vector along any line /. 
The projection of a along / is a/1; see 4*05. Any vector is the sum of 
its projections along three perpendicular axes; see 4*06. So a is ex- 
pressed in natural coordinates as 



a = a,t -f 
and in standard coordinates as 

a - a x i + a v j 

The vector a may also be projected into any plane. This projection 
is a vector equal to the sum of the two projections of a along any two 
perpendicular lines in the plane. Let the vector projection of a in the 
horizontal plane be denoted by a# . This vector is then given by 

a// - a 8 t -f- a n n = a x i + a y j. 

Moreover, the vector a is equal to the sum of its projection in a plane 
and its projection along the normal to that plane. Thus 

a - ajy + a z k. 

The equation of relative motion 
6-19(3) v = b+c + g 

expresses Newton's second law: The observed relative acceleration 
equals the resultant of all acting forces, including inertial forces due to 
the motion of the relative frame. The vectors in the equation of rela- 
tive motion can be projected along any line or into any plane. The pro- 
jected equation expresses Newton's second law along that line or in that 
plane: The relative acceleration projection equals the sum of all the 
force projections. 



175 Section 7-05 

Therefore the equation of relative motion projected along the line / is 

(1) vi~bi+ci + gi. 

And the equation of relative motion projected into the horizontal plane is 

(2) V H = b// + c//. 

Here g# does not appear, for the force of gravity has no horizontal 
component. 

704. The acceleration. The acceleration of a particle in arbitrary 
motion is 

6-05(9) v = i>t+v 2 K. 

The vector curvature, directed toward the center of curvature and nor- 
mal to the unit tangent, is then given by 

(1) K = # B n + * z k. 
Therefore the acceleration becomes 

v = vt + v 2 K n a. + v 2 KJt. 
The natural components for horizontal flow are then 

(2) *. = *, 

(3) i> n = v*K n , 

(4) *. = v 2 K t . 

The horizontal and vertical components of the centripetal acceleration 
contain respectively the horizontal and vertical components of the vector 
curvature. These curvature components will now be examined in detail. 

7*05, Cyclic sense. When the motion of a particle is horizontal, the 
path of that particle must lie in a level surface. Although a level surface 
is not exactly spherical (section 6-10), it may be considered spherical in 
so far as the geometry of the path is concerned. 

The intersection between a sphere and an arbitrary plane is always a 
circle. If the plane passes through the center of the sphere, the circle is 
called a great circle. Otherwise, the circle is called a small circle. The 
circle cut by the osculating plane at any point of a spherical path is the 
circle of curvature at that point. Its axis of rotation is perpendicular 
to the osculating plane at the center of curvature and passes through the 
center of the earth. 

Momentarily a particle moves along the circle of curvature about the 
axis of rotation. The particle may move with either cyclic sense along 
this circle. If the circle of curvature is a small circle, the cyclic sense 



Section 7-05 



176 



will be defined as positive when the particle appears from the local zenith 
to be moving counterclockwise, and as negative when the particle appears 
to be moving clockwise. This definition is illustrated in the left-hand 
(c + ) and right-hand (c~~) diagrams of fig. 7-05. The lower diagrams are 




FIG. 7-05. Cyclic sense. 

in the plane perpendicular to the path and show the particle at P moving 
away from the reader. And the upper diagrams show the circle of 
curvature projected on the horizontal plane as it appears from the zenith 
at P. 

If the circle of curvature of a particle is a great circle, the osculating 
plane passes through the local vertical. Consequently the path of the 
particle appears from the local zenith to be straight, as shown in the 
center diagram (g) of fig. 7*05. Flow momentarily along a great circle 
will be called geostrophic. Evidently geostrophic flow represents the 
transition between the positive and negative cyclic senses. So the 
cyclic sense of geostrophic flow may be defined either as positive or as 
negative. 

In section 6-05 the quantities ^, K, R, co were defined as positive. 
However, for the discussion of horizontal flow it will be convenient to 
assign to them the cyclic sense as defined above. Accordingly in many 
of the equations of section 6-05 the symbols ^, K, R, o> must be replaced 
by their absolute values. But this is not necessary in some equations, 
namely, in 6-05(5, 6, 7, 8). 

Notice that the cyclic sense of rotation of a particle fixed to the earth is 
positive in the northern hemisphere and negative in the southern hemi- 
sphere. So 12 is positive in the northern hemisphere and negative in the 
southern hemisphere. For this reason the cyclic sense of zonal flow in 



177 



Section 7-06 



the southern hemisphere was not defined in section 6-11. However, all 
the equations of zonal flow so far introduced are valid in both hemi- 
spheres for either cyclic sense of rotation. 

Internal consistency within this chapter requires that 12 have sense. 
That is, if the sense of angular rotation is distinguished by assigning 
cyclic sense to angular speeds, then, in particular, cyclic sense must be 
assigned to the angular speed of the earth. Accordingly the definitions 
and equations of this chapter apply to any horizontal flow in either hemi- 
sphere. However, since it is customary to regard 12 as positive, we shall 
strictly observe all the consequences of the cyclic sense convention in 
this chapter only. 

7-06. Angular radius of curvature. Let 6 be the acute angle (^ ?r/2) 
between the osculating plane and the horizontal plane. The angle 6 
will be called the angular radius of curvature, for, as shown in fig. 706a, 

z 





FIG. 7-06a. Angular radius of curva- 
ture. 



FIG. 7-06&. Nat ural components of K. 



it is the angle subtended at the center of the earth by the radius of curva- 
ture. In particular, the angular radius of curvature for zonal flow is the 
colatitude. 

The atmosphere may be considered as a thin blanket enveloping the 
earth. Whenever atmospheric flow is investigated, the distance to an 
air particle from the center of the earth is nearly equal to the mean 
radius of the earth. Consequently the radius of curvature subtending 6 
is R = a sin 0, and its reciprocal, the curvature, is 



This equation is valid for flow with either cyclic sense when we adopt the 
convention that 6 has the same sign as the cyclic sense. 



Section 7-06 



178 



The vector curvature appears from the zenith to point to the left of 
flow with positive cyclic sense, and to the right of flow with negative 
cyclic sense. (See fig. 7-05.) The horizontal unit normal always points 
to the left of the flow, so the normal component K n = K*n of the vector 
curvature has the same sign as the cyclic sense. It is seen from fig. 7-066 
that the normal component of the vector curvature is 



(2) 



K n = K COS 0. 



Evidently this equation is valid for flow with either cyclic sense. 

Since the center of curvature lies below the horizontal plane, the 
vertical component K z of the vector curvature is always negative. So, 
as shown in fig. 7-066, this vertical component is 

(3) K z = -K sin 0. 

The curvature K may be eliminated from the expressions for K n and 
K a . When (1) is substituted in (2) and (3), we find 

1 



(4) 



(5) 



a tan 

_ 1 
a 



We shall now show that these projections of the vector curvature of the 
spherical path are the curvatures of the corresponding projections of the 
spherical path. 

7-07. Horizontal curvature. When the spherical path of a particle 
is projected onto the horizontal plane, a curve called the horizontal path 

is obtained. The curvature of this 
horizontal path is called the horizontal 
curvature and will be denoted by K/f. 
Consider an arc of an arbitrary 
spherical path c near the point P; see 
fig. 7-07a. This arc may be thought 
of as being in the osculating plane w 
of the spherical path at P. When 
the path c is projected from w onto 
the horizontal plane TT H at P, we 
obtain the horizontal path CH- The 
line of intersection between the planes 
TT and wfj is the line PAA\, tan- 
gent at the point P both to c and CH. The two curves c and C H define 
a cylinder normal to the horizontal plane TT#, whose generators are the 




Horizontal curvature. 



179 Section 7-07 

lines of projection from c to CH- The plane tangent to this cylinder along 
the generator P\P\n intersects the tangent line at the point A. And 
the plane through the generator PI PI// normal to the line PAAi inter- 
sects the line PAAi at AI. The angle PI A \P\n is then the angle 6 
between the osculating plane and the horizontal plane. 
The curvature of the spherical path c at P is by definition 

6-05(5) #=^> 

as 



where dfy = ^ AiAPi is the angular turn of the tangent to c through the 
infinitesimal arc length ds = PP\. Similarly the curvature of the hori- 
zontal path CH at P is 



(^^ v 

(1) K H = - > 

ds n 

where d\f/H = ^ A\AP\n is the angular turn of the tangent to CH 
through the arc length dsn = PPi//. 

As Pi,Pi//->P, the arc length ds along c and dsn along CH become 
equal, so in the limit 



- - 

dsn PPi n ' 

And as PI, PI// > P the angles df, d^n have the limiting ratio 

ir _ ^iPi_?/ _ 

~ "~ " OS " 



d$ " AA l 

Therefore the horizontal curvature is given by 

ds 



K H == = -77 ~T 3^ = cos 
ai^ a5 as// 



And finally from 7-06(2) we find 

(3) #// = K cos 



Consequently, /Ae curvature of the horizontal projection of the path is equal 
to the horizontal component of the vector curvature. So, from 7-06(4), the 
horizontal curvature may be expressed in terms of the angular radius of 
curvature as 

(4) KH = -- 

W H a tan 8 

Evidently the horizontal path is less strongly curved than the spherical 
path. The horizontal projection of the circle of curvature of a spherical 



Section 7-07 



180 



path is an ellipse (see fig. 7-05). The horizontal curvature is the curva- 
ture of this ellipse at the point P. However, the horizontal curvature 
may also be considered as the curvature of a circle, the horizontal circle 
of curvature, whose radius RH is the reciprocal of KH- This radius is 
called the radius of horizontal curvature and is given by 

RH = a tan 6. 

Notice that the smallest numerical value of the space curvature for 
horizontal flow is the curvature of a great circle. Since an arc of a great 
circle projects into a straight line on the horizontal plane, the horizontal 
curvature of a great circle is zero. Hence geostrophic flow, which is 

great circle flow on the earth, projects 
as linear flow in the horizontal plane. 
For this reason the cyclic sense of geo- 
strophic flow is unimportant. 

The horizontal curvature has consider- 
able practical importance because the 
meteorologist must work with plane maps 
of the earth. The curvature of a curve 
represented on a plane map of the earth is 
more nearly given by the horizontal cur- 
vature than by the space curvature, for 
a good map of the earth combines the 
horizontal plane projections at every 
point of the mapped region into a whole 
map with as little distortion as pos- 
sible. 



H 




FIG. 7-076. 



Vertical component 
of w. 



The horizontal curvature is closely associated with the component co^ 
of the angular velocity about the local zenith. Fig. 7-07& shows that this 
component is 

(5) CO;5=COCOS0. 

Since, by (3), KH = K cos0, the relation 



6-05(8) 

may be expressed as 

(6) 



,--, 



V = 



Thus, as a particle moves along a spherical path with the angular speed 
co, its projection moves along the horizontal path with the angular 
speed co z . 



181 



Section 7-08 



7*08. Geodesic curvature. Let PI and P% be any two distinct points on a 
surface cr. The shortest curve in <r joining PI and P% is a geodesic of cr. Physi- 
cally this geodesic can be visualized as the curve taken by a taut thread 
stretched between P\ and P% on the convex side of a shell having the surface <r. 
If a is planar, the geodesies of a are straight line segments. And if cr is spheri- 
cal the geodesies of cr are great circle arcs. 

The geodesic curvature K Q of a curve c on a is defined as 

K d +", 
K " ~' 



where d\l/ a is the angular turn of the geodesic, tangent to the curve, through the 
arc length ds. The construction of the angular turn between the geodesies g 




FIG. 7-08a. 

and gi, tangent to c at P and PI respectively, is illustrated in fig. 7-08a for a 
plane curve, and in fig. 7-086 for a spherical curve. Notice that the geodesic 
curvature is a generalized concept of curvature defined by an operation 
entirely on the given surface. 




FIG. 7-086. 

If <r is planar, evidently the geodesic curvature is the ordinary curvature. 
But if a is curved, the geodesic and ordinary curvatures are different. We 
shall now show that the geodesic curvature of a spherical curve is the horizon- 
tal curvature. Such a brief treatment will naturally be incomplete. 

The spherical geodesies g and g\, represented in fig. 7-08c, are great circles 
whose planes intersect the horizontal plane tangent at P in the straight lines 



Section 



182 



gn and gin respectively. On the spherical surface g and g\ meet at A, and on 
the horizontal plane gn and g\n meet at AH- As PI JP, we see that A,A n*P 
and thatd\l/ff,d\l/ >0. However, in the limit, 



= 1. 



By 7-07(1, 2) we finally obtain 



ds 



d\l/n dsu ds 

Consequently the geodesic curvature of a spherical curve is equal to its horizontal 
curvature. The curvature ///, which occurs so often in the description of 
horizontal flow, may then be considered either as the geodesic curvature of the 
spherical path or as the planar curvature of the horizontal projection of the 
spherical path. 




FIG. 7-08c. 

709. Vertical curvature. Let the plane through the unit tangent 
and normal to the horizontal plane be called the vertical plane. When 
the spherical path is projected onto this vertical plane we obtain a curve 
called the vertical path whose curvature is the vertical curvature. 

If the argument of section 7-07 is applied to the vertical path, we find 
that the curvature of the vertical path is equal to the vertical component 
of the vector curvature. Therefore the vertical curvature is K z . Its 
reciprocal, the radius of vertical curvature, will be denoted by R z . From 
7-06(5) this radius is 

R z = a. 

Since a is the radius of a great circle, the spherical path projects into the 
vertical plane as an arc of a great circle no matter how strongly curved the 
spherical path may be. Again we see how important is the notion of the 
great circle in horizontal flow. 

In fig. 7-09 the three radii of curvature R, RH, and R z are drawn to the 



183 



Section 7-10 



moving point P from the centers of curvature C, C#, and C z respectively. 
These centers are collinear points on the axis of rotation. The angular 
radius of curvature is then subtended both by R and 



H 




FIG. 7-09. 

Since the horizontal and vertical components of the vector curvature 
represent curvatures, the horizontal and vertical components of the 
centripetal acceleration represent centripetal accelerations. As a parti- 
cle moves along a spherical path, its projection in the horizontal plane 
moves along the horizontal path with the centripetal acceleration Kuv 2 , 
and its projection into the vertical plane moves along the vertical path 
with the centripetal acceleration K z v 2 . 

7*10. The angular velocity of the earth. The angular velocity of the 
earth is a vector parallel to the axis of the earth and extending northward. 
Since the natural coordinate system coin- 
cides with the standard coordinate system 
for a particle fixed to the earth, the simplest 
decomposition of Q is afforded by standard 
coordinates. Moreover, we shall find that 
the natural coordinate components of fl are 
not required. 

Since the angular velocity of the earth is a 
meridional vector, 

(1) fi x =0. Equator 

The other standard components are in the FlG - Ma Rectangular corn- 
meridional plane. Fig. 7-10 shows that ponentsofo. 
these components are easily expressed in terms of the latitude <p. The 
component to the local north is 

(2) n,-|i 



Pole 




Section 7-10 



184 



and the component along the local vertical is 

(3) Q f -|n|sin?. 

The vertical component of the earth's rotation is positive in the northern 
hemisphere and negative in the southern hemisphere. Equation (3) 
is valid in both hemispheres when the latitude angle tp is considered posi- 
tive in the northern hemisphere and negative in the southern hemisphere. 
Since 12 is positive in the northern hemisphere and negative in the 
southern hemisphere, the component of the earth's rotation about the 
local zenith may also be written 



(4) 



12 2 = 12 sin \<p\. 



Evidently equation (2) is not altered by the sign of the latitude. The 
component of to the local north is always positive. 

The components of the angular velocity of the earth are required for 
the decomposition of the Coriolis force. 

711. The Coriolis force. The Coriolis force is given by the vector 
product 

6-19(2) c=-2flxv. 

We shall evaluate this vector product in standard and natural compo- 
nents by using the determinant expression 6-13 (6). 

The Coriolis force may then be expressed in standard components by 
the determinant: 

j k 



-2 











Expansion of this determinant gives 

c = 212^1 - 212>J + 2!2yifek. 
So the standard components of the Coriolis force are: 

(1) ^=212^, 

(2) c v =-2Q z v x , 

(3) c z = 2Q v v x . 

And the Coriolis force may be expressed in natural components by the 
determinant: 

t n k 



-2 



12, 12 n 
v 



185 Section 7-12 

Expansion of this determinant gives 



So the natural components of the Coriolis force are: 

(4) c*=0, 

(5) c n 

(6) c z 



Since the Coriolis force is normal to the velocity, the tangential com- 
ponent c a is zero. It is for this reason that the Coriolis force has been 
called the deflecting force. 

The normal component c n is then the only horizontal component. The 
vertical component of the earth's angular velocity occurring in the ex- 
pression for c n is positive in the northern hemisphere, zero at the 
equator, and negative in the southern hemisphere. Accordingly the hor- 
izontal Coriolis force as seen from the zenith acts to the right in the 
northern hemisphere, is zero at the equator, and acts to the left in the 
southern hemisphere. 

Since the only horizontal component of the Coriolis force is normal 
to the flow, the horizontal vector component c// is given by 



We shall define the vector fl z , extending along the local vertical, by 



Therefore the horizontal vector component of the Coriolis force may be 
written in the form 

(7) c# = -2ti z vn = -2^ z v\a xt = -2Q Z xv. 

Finally, the vertical component c z is given either by (3) or by (6). 
Both expressions are, of course, the same. However the standard com- 
ponent form is more convenient to use. 

7- 12. The pressure force and the force of gravity. So far the com- 
ponents of two vectors in the equation of relative motion have been 
found. The other two vectors are the force of gravity and the pressure 
force. These forces are easily resolved. 

The components of the force of gravity have already been obtained in 
442 (6) . This force has no horizontal components, so g a , g n , gx, gy are all 
zero. The only component of the force of gravity is in the vertical, so 
gz - g*k - -g. 

The components of the pressure force may be obtained by reference to 



Section 7-12 



186 



section 4-15. When equation 4-15(4) is multiplied by the specific vol- 
ume, we find for the component of the pressure force in the direction / 

bi = aWp = a > 

and the horizontal pressure force is 

(1) bff = -otVnP- 

This horizontal force acts normal to the horizontal isobars toward lower 
pressure. 

7-13. The component equations of relative motion. We have 
examined the standard and natural components of each vector in the 
equation of relative motion. The component equation along any line / 
is 

7-03(1) vi = bi + ci -f- gi. 

This equation shows that the relative acceleration component equals the 
sum of the force components. The components i>i, 6j, ci, gi have been 
obtained in the previous sections. They are assembled in the following 
table. 



/ 1 


vi bi d 


gi 


X 


i 


dp 





y 


j 


dp 

Vy OL 2il z V x 





z 


k 


Vg ot ~ 2i2yV x 


-f 


s 


t 


dp n 
v a 

ds 





n 


n 


2 9/> 





z 


k 


v 2 dp 








a dz 





The standard component equations are given by the first three rows 
of the table. They are 



(1) 



-a 



187 Section 7-14 

d/> 

(2) !>=- ^- 

dp 

(3) t>* --a -^+ 

These equations are displayed here for reference. They are often used 
in the investigation of atmospheric dynamics when it is desired to have 
the orientation of the coordinate system independent of the motion 
(see chapter 12). 

However, we shall examine in detail only the natural component equa- 
tions, given in the last three rows of the table. They are 

(4) , , |, 

(5) K H v 2 = -<x^-2n z v, 

on 

v 2 d 

(6) --=_- +2Qj> x -g. 

a oz 

These component equations along the natural coordinates 5, n, and z 
are called the tangential, normal, and vertical equations respectively. 
They will be discussed in the following sections. 

7 14. The vertical equation. The vertical equation of relative 
motion equates the vertical centripetal acceleration to the sum of the 
vertical components of the acting forces. Thus, 

v z = b, + c z - g. 

The force of gravity occurring in this equation is the resultant vertical 
force measured by an observer fixed to the earth. However, if the 
observer moves horizontally relative to the earth the resultant vertical 
force which he measures is not equal to the force of gravity, for it also 
includes the vertical inertial forces due to the motion, namely, the verti- 
cal centrifugal force and the vertical Coriolis force. The resultant verti- 
cal force will be called the virtual gravity. Its magnitude g* is given by 

v 2 



A moving particle will be called " heavier," if g* > g, and " lighter," if 



*< 



The vertical centrifugal force is the centrifugal force which would be 
exerted on the projection of the particle moving along the vertical path..? 



Section 7-14 188 

Since the vertical path is an arc of a great circle, the effect of the vertical 
centrifugal force opposes the force of gravity, always making the particle 
lighter. 

The vertical Coriolis force acts upward when the particle moves 
toward the east and downward when the particle moves toward the west. 
So its effect is to make the particle lighter for eastward motion and heav- 
ier for westward motion. Moreover, for a given horizontal velocity the 
effect of the vertical Goriolis force, being proportional to cos <p, is zero at 
the poles and greatest at the equator. 

Even for the strongest flow speeds (100 m s" 1 ) attained by the atmos- 
phere the centrifugal and Coriolis correction terms to the force of grav- 
ity are small. The greatest error occurs in eastward flow at the equator. 
For v = 100 m s" 1 the centrifugal correction is then 0.0016 m s~ 2 . And 
the Coriolis correction is 0.0146 m s~ 2 . So the total error is 
0.0162 m s~ 2 . When this error is compared with the mean value 
g = 9.81 m s~~ 2 , we find that the correction terms are usually negligible 
for horizontal atmospheric flow. Hence g* may usually be replaced by g. 
That is, 

(1) g*~g. 

When the vertical equation is expressed in terms of the virtual grav- 
ity, we find 

(2) b, = g*. 

The vertical pressure force is then completely balanced by the force of 
virtual gravity. (2) is called the generalized hydrostatic equation. 
When the relative flow is zero, g* = g, and (2) becomes the hydrostatic 
equation. 

For an atmosphere in horizontal motion height should be computed by 
the generalized hydrostatic equation. However, by (1) the hydro- 
static equation is approximately satisfied. In fact, the error introduced 
by neglecting the inert ial contributions to the virtual gravity is less than 
the unavoidable instrumental errors of the sounding equipment. There- 
fore the practical use of the hydrostatic equation in chapter 4 for com- 
putation of height is justified whenever the atmospheric flow is hori- 
zontal. 

715. The tangential equation. The tangential equation of relative 
motion equates the change of speed to the tangential component of the 
pressure force. Thus 

(1) *-&.- -a^- 



189 Section 7-15 

A particle moving horizontally changes speed only when crossing hori- 
zontal isobars. A particle moving toward lower pressure speeds up, and 
a particle moving toward higher pressure slows down. Since a particle 
moving along a horizontal isobar is not subject to a tangential pressure 
force, it moves momentarily with constant speed. Horizontal flow 
along the isobars, and therefore normal to the pressure gradient, is called 
gradient flow. Gradient flow, accordingly, occurs at points where 
v = 0, that is, at points of constant or extreme (maximum, minimum) 
speed. 

If the flow at every point of the atmosphere were gradient flow, the 
horizontal isobars would everywhere be tangent to the wind direction. 
But in general the atmospheric flow is gradient only at isolated points. 
However, at any instant curves can be drawn which are everywhere 
tangent to the wind direction. Such curves are called streamlines. The 
streamlines give a snapshot of the flow direction throughout the entire 
atmosphere at a fixed time; they represent the flow pattern at that time. 
Usually this flow pattern varies from instant to instant. If the flow 
pattern is the same at every instant, the flow is called steady. The 
flow direction at every point is then independent of time. 

We must make a clear distinction between a streamline and a path. 
A streamline is a curve tangent to the velocities of different air particles 
at a given instant, whereas a path is a curve tangent to the velocities of a 
given air particle at different instants. At a fixed time a streamline can 
be drawn through any given point of the atmosphere. Through the 
same point a path can also be drawn, namely, the path of the air particle 
which momentarily occupies that point. Since these two curves, the 
path and the streamline, have the same direction at the given point, they 
are tangent curves, but in general the two curves have different shapes. 
If the flow is steady, the path and the streamline through the given point 
coincide. 

In the atmosphere the flow patterns are usually changing, so the 
streamlines and the paths have different shapes. However, horizontal 
currents in the atmosphere change speed rather slowly. When the 
current speeds up it flows slightly across the isobars toward lower 
pressure, and when the current slows down it flows slightly across the 
isobars toward higher pressure. The streamlines are therefore nearly 
along the isobars. This practical rule allows the horizontal pressure 
field to be drawn with considerable accuracy from a few scattered wind 
and pressure data. The relation between the streamlines and the iso- 
bars will be examined in more detail in chapter 12. 

This discussion holds only for horizontal flow in the free atmosphere 
above the frictional layer. Near the surface of the earth the motion has 



Section 7-15 190 

a component toward lower pressure as a consequence of friction. The 
flow under frictional forces will be treated in chapter 9. 

716. The normal equation. The normal equation of relative motion 
equates the horizontal centripetal acceleration to the resultant of the 
normal pressure force and the horizontal Coriolis force. Thus 

(1) v n =b n +c n . 

Although this equation clearly shows the dynamics of the flow, the kine- 
matics is better understood when v n and c n are expressed in terms of the 
speed. We then obtain from (1) 

(2) K n v 2 +2tt z v- b n = 0. 



This equation is quadratic in v. Since il z = |fl| sin <p, the speed depends 
on the horizontal curvature, the latitude, and the normal pressure force. 

The latitude is by our convention (see section 7*10) positive in the 
northern hemisphere and negative in the southern hemisphere. The 
two latitudes (^, -^>) will be called corresponding latitudes. At corre- 
sponding latitudes the normal equation (2) gives the same speed v for <p, 
KH, b n in the northern hemisphere as for <p, Kit, b n in the southern 
hemisphere. At corresponding latitudes, (Kn> Kn) will be called 
corresponding horizontal curvatures, and (b n , -b n ) will be called corre- 
sponding normal pressure forces. 

These definitions of correspondence are equivalent to replacing n in 
the northern hemisphere by n in the southern hemisphere. That is, 
the direction to the left of the flow in the northern hemisphere corresponds to 
the direction to the right of the flow in the southern hemisphere. The physi- 
cal reason for this interchange is clear; the horizontal Coriolis force acts 
to the right of the flow in the northern hemisphere and to the left of the 
flow in the southern hemisphere. In corresponding flow the normal pres- 
sure force and the horizontal centripetal acceleration in both hemispheres 
have the same orientation with respect to the horizontal Coriolis force. 

If the normal pressure force is opposite to the horizontal Coriolis 
force, the flow is called baric; and if the normal pressure force is along the 
horizontal Coriolis force, the flow is called antibaric (see fig. 7-16). In 
the northern hemisphere the flow is baric if low pressure lies to the left 
of the flow, and antibaric if low pressure lies to the right. In the south- 
ern hemisphere the flow is baric if low pressure lies to the right of the 
flow, and antibaric if low pressure lies to the left. Finally, if the normal 
pressure force is zero the flow is called inertial, for the only horizontal 
force, the horizontal Coriolis force, is an inertial force. 

If the horizontal centripetal acceleration is opposite to the horizontal 
Coriolis force, the flow is called cyclonic-, and if the horizontal centripetal 



191 



Section 7-16 



acceleration is along the horizontal Coriolis force, the flow is called anti- 
cyclonic (see fig. 7-16). In the northern hemisphere flow curved to the 
left is cyclonic and flow curved to the right is anticyclonic. In the south- 
ern hemisphere flow curved to the right is cyclonic and flow curved to 
the left is anticyclonic. Finally, if the horizontal centripetal accelera- 
tion is zero the flow is gcostrophic. 

The above definition of cyclonic and anticyclonic sense generalizes to 
arbitrary horizontal flow the definition given earlier for zonal flow. 
This generalization is more clearly exhibited by the following equivalent 
definition of cyclonic and anticyclonic sense, based upon the sense of the 



Baric 



H 



Northern hemisphere 
Antibaric Cyclonic 



I; 

H 



in 



Anticyclonic 



Equator 



H 



Baric 



Antibaric Cyclonic 
Southern hemisphere 

FIG. 7-16. 



Anticyclonic 



earth's rotation about the local zenith. Flow which appears from the 
local zenith to have the same sense as the rotation of the earth about that zenith 
is cyclonic, and flow which appears to have the opposite sense is anticyclonic. 
Both this definition and the definitions of positive (counterclockwise) 
and negative (clockwise) cyclic sense refer to the flow as it appears from 
the local zenith. That is, these definitions refer to the horizontal path 
of the flow. Cyclic sense is defined according to an arbitrary rule, 
whereas cyclonic sense and anticyclonic sense are defined according to 
the sense of a physically given rotation the rotation of the earth about 
the local zenith. 

Cyclonic flow and anticyclonic flow are defined according to the orien- 
tation of the horizontal centripetal acceleration with respect to the hori- 
zontal Coriolis force. Baric flow and antibaric flow are defined accord- 
ing to the orientation of the normal pressure force with respect to the 
horizontal Coriolis force. These definitions do not apply at the equator 



Section 7-16 192 

where the horizontal Coriolis force is zero. However, even at the equa- 
tor the definitions of geostrophic flow (K H = 0) and of inertial flow 
(b n 0) apply. 

Before examining the normal equation for arbitrary horizontal flow, 
we shall consider the three simple but physically important flow types 
which occur when one term of the normal equation is zero. 

7-17. Geostrophic flow. Flow along a great circle is geostrophic. 
The horizontal curvature and the horizontal centripetal acceleration are 
then zero, so the normal pressure force is completely balanced by the 
horizontal Coriolis force. The two forces are equal and opposite, and 
the normal equation becomes 

(1) 0=J n + c n . 

Since the normal pressure force is opposite to the horizontal Coriolis 
force, geostrophic flow is baric. That is, a geostrophic current has low 
pressure to the left in the northern hemisphere and low pressure to the 
right in the southern hemisphere. This rule, known as the baric wind 
law, was discovered empirically by Buys Ballot in 1857. 

The geostrophic wind speed will be denoted by v g . Solving equation 
(1) for this speed, we find 



Notice that geostrophic flow is the same for corresponding normal pres- 
sure forces at corresponding latitudes. 

The dependence of the geostrophic wind upon the normal pressure 
force and the latitude is shown graphically in fig. 7-1 7a. Isopleths of v g 
are plotted against linear coordinates of b n and $l g . The v g isopleths are 
straight lines radiating from the point b n = 0, <p = off the graph. Since 
the geostrophic wind becomes infinite at the equator for any non-zero 
normal pressure force, geostrophic flow cannot be realized in equatorial 
regions. So the graph is cut off at <f> = 30. 

To evaluate the normal pressure force the horizontal pressure field 
must be known. Let the horizontal pressure field be represented by 
horizontal isobars drawn for the pressure interval A. The variation of 
pressure in the n direction may be found by measuring the distance An 
normal to the flow between consecutive isobars. 

The most convenient unit of distance on a weather map is the degree 
of latitude. This unit is independent of the map projection. If the 
earth is regarded as a sphere of radius a, the degree of latitude has the 
constant length of Tra/180 - 111.1 km. See 1-04(1). 

Let the distance Aw, expressed in degrees of latitude, be H n . Then 



193 



Section 7-17 



Aw = 1.11 x W 5 H n . If H n is measured toward lower pressure between 
isobars drawn for 5-mb intervals, then A = -0.5, and 



(3) 



a 



,- -a ^=4.50X10-"-^ 
Aw H n 



10 15 20 25 30 35 40 




2015 10 8 7 6 5 4 

-Hn v 



5 

700f3(> 



800 
900 
1000 
1100 
1200 
1300 
1400 



2.5 



10 15 20 25 30 35 40 




2015 10 8 7 6 5 



Top: FIG. 7-17a; middle: FIG. 7-176; bottom: FIG. 7-17c. 

The graphical solution of this equation is shown in fig. 7-176. Isopleths 
of H n are plotted against linear coordinates of b n and a. The H n iso- 



Section 7-17 



194 



pleths are straight lines radiating from the point b n = 0, a = off the 
graph. 
When b n is eliminated from (2, 3), we find 



(4) 



This elimination of b n may be performed graphically in two operations: 
the first on fig. 7-1 la and the second on fig. 7-176. When these figures 
are superposed, the v g isopleths and the H n isopleths radiate from the 
same point. So both sets of isopleths may be represented by one set of 
lines. Fig. 7-17c shows the superposed graphs with one set of lines. 
These lines are drawn for integral values of II n . They are labeled for 
II n along the bottom and right side. And they are labeled for v g along 
the top. The left side serves both as an a coordinate and a <p coordinate. 
Fig. 7-17c gives v g = v g (a,H n ,<p). a is not directly available from 
meteorological observations. From the equation of state we have 



4-17(3) 



a = 



W 

80, 



iio: 



This equation is represented graphically in fig. 7-l7d. Isopleths of p 
are plotted against linear coordinates of a and T*. The p isopleths are 

straight lines radiating from the 
point a. = 0, T* = off the graph. 

The complete practical solution 
v g = v g (p,T*JI n ,<p) is obtained by 
joining fig. 7-1 7d to the left side 
of fig. 7-1 7*;. The resulting graph 
is shown in fig. 7-1 7e, and an ex- 
ample of the procedure is illus- 
trated in fig. 7-17/. From the data 

^ p = 100 cb, T* = 0C, H n - 3, p = 

-50-40-30-20-10 10 20 30 45, the value v g = 11.4 m s" 1 is ob- 
r*(C) tained. The elimination of b n is 

* TJ ^ t. r r shown in this diagram by the line 

FIG. 7-17o. Graph of equation of state. , u f , u , ,, 

labeled b n = const connecting 

the operations in the two superposed graphs. 

In practical application the value of the speed is not so much required 
as the displacement of an air parcel from one weather map to the map 
twelve hours later. The unit of distance is again the degree of latitude. 
The total twelve-hour displacement in degrees of latitude is denoted by 
D. If the speed is constant throughout the twelve-hour interval 



7001 

800 

900 

1000 

1100 

1200 

1300 
1400 




50 



195 



-Section 7-17 




FIG. 7-17e. 



FIG. 7-17/. 



Section 7- 17 196 

between maps, D is merely a constant multiple of v. Thus 



Even though the current speeds up or slows down, the value defined by 
(5) can be calculated. Evidently this value of D is often more useful 
than is the value of v. A scale of D has been placed along the v g scale in 
fig. 7-1 7e. The displacement in the above example is D = 4.4. 

7-18. Inertial flow. When the pressure force has no normal com- 
ponent, the horizontal flow is inertial. The horizontal centripetal 
acceleration is then equal to the horizontal Coriolis force, and the normal 
equation becomes 

(1) v n =c n . 

Both the centripetal acceleration and the Coriolis force are zero for a 
resting particle, so (1) is satisfied by 

(2) IF-O. 

However, (1) also has a non-zero solution which will be denoted by v*. 
This non-zero inertial wind speed is then given by 

(3) - 

Since the horizontal centripetal acceleration is along the horizontal 
Coriolis force, inertial flow in both hemispheres is anticyclonic. An 
inertial current crossing the equator must then change cyclic sense. So 
inertial flow at the equator is geostrophic. 

The graphical solution of (3) is shown in fig. 7-18a. Isopleths of v 
are plotted against linear coordinates of Q 2 and KH. The Kn coordi- 
nate is labeled both in multiples of the curvature of a great circle and in 
degrees of angular radius of curvature. The v* isopleths are straight 
lines radiating from <p = 0, KH = 0. They intersect any coordinate of 
constant latitude in a reciprocal Vi scale. This fact will be useful later. 

For inertial flow the angular speed of the horizontal projection of a 
particle is, from (3) and 7-07(6), 



(4) u zi 

And the angular speed of inertial flow along a spherical path is, from 
7-07(5), 

(5) = ****' a - 2 * 

W * ~~ cos e ~ cos ' 



197 



Section 7- 18 



When the flow is zonal, the angular radius of curvature is the colatitude. 
Therefore cos 6 = sin |^|, and (5) becomes w z - = -2Q by 7-10(4). From 
(2) , the angular speed of inertial flow may also be zero. Both results are 
verified in fig. 6'llc 2 ,2 Inertial zonal flow occurs only when the iso- 
baric and level surfaces coincide. 

Inertial flow can be realized momentarily by a horizontal current 
flowing directly across horizontal isobars toward lower pressure. This 



20 
15 

q 

| 

10 



2.5 

3 
3.5 



10 



20 



30 



40 



8 
10 

15 
20 



90 




50 

60 

70 

80 5 

90 

100 



200 



5 10 15 20 25 30 35 40 45 50 55 60 70 90 
V 

FIG. 7-18a. Graph of inertial wind speed. 



current would curve anticyclonically and become baric. It would also 
speed up, for the pressure force at the moment of inertial flow would be 
wholly tangential. 

However, as in zonal motion, inertial flow over an extended region can 
be realized only when an isobaric surface is level. Both the normal and 
the tangential pressure forces are then zero. And a current will move 
anticyclonically with constant speed in an inertial path. Since the 
magnitude of the earth's rotation about the local zenith is greater near 
the poles and zero at the equator, the curvature of the inertial path is 
also greater near the poles and zero at the equator. 

An inertial path crossing the equator will oscillate about the equator 
between the corresponding latitudes <p, <p, as shown in fig. 7486. An 
inertial path not crossing the equator will oscillate between two lati- 
tudes <p P , <p Ej as shown in fig. 7-18c. The path will be looped, for the 



Section 7-18 198 

curvature is greatest at the latitude <p p nearest the pole and least at the 
latitude tp E nearest the equator. 

As z; t -->0, the inertial path becomes more strongly curved, and the 
inertial circle of curvature shrinks to a point. The limiting inertial 
period r^ required for the particle to rotate about this inertial circle 
centered at the latitude <p is given by 

2ir 2ir 1 sidereal day 

(6) Ti - : : - - -,-r-T = r-r-j 

|co 2i -| 2\\lg\ 2 sin \<p\ 

The particle completes one loop of the inertial path approximately in the 
inertial period r,-. 

n n 





FIG. 7-186. FIG. 7-18c. 

The inertial period is simply related to the period of relative revolu- 
tion of the plane of oscillation of a freely suspended pendulum. This 
plane maintains its orientation in space, while the earth rotates under it 
with the angular speed $l z . It appears from the earth as if the pendulum 
plane rotates about the vertical in the opposite direction. The period 
of revolution of the pendulum plane is called the pendulum day, and is 
given by 

27T sidereal day 

~~ 



Evidently r p = 2r, so the inertial period is one-half of the pendulum day. 

7-19. Cyclostrophic flow. Whenever horizontal flow in the northern 
hemisphere is compared with horizontal flow in the southern hemisphere, 
the direction of the horizontal Coriolis force and the sense of the earth's 
rotation about the local zenith are important. At the equator both the 
horizontal Coriolis force and the earth's rotation about the zenith are 
zero, so the normal equation becomes 

(i) * - &. 



199 



Section 7-20 



Unless the normal pressure force is zero, equatorial flow cannot be geo- 
strophic. The circle of curvature is then a small circle, and the flow is 
called cyclostrophic. By (1) the center of horizontal curvature of a 
cyclostrophic current lies on the low-pressure side of the current. 
When (1) is solved for the cyclostrophic wind speed v c we find 

z> 

(2) 



' K H 

This equation is represented graphically in fig. 7-19. Isopleths of v c 
are plotted against linear coordinates of b n and KH- These isopleths are 
straight lines radiating from the point b n = 0, KH = 0. The normal 
pressure force b n may be evaluated from the graphs for geostrophic flow. 



20 



15 



10 



2.51 



10 




OJ-90. 



FIG. 7'19. Graph of cyclostrophic wind. 

720. Arbitrary horizontal flow. The normal equation of horizontal 
motion is 

7-16(1) v n =b n + c n . 

We have examined this equation when one of the three terms is zero. In 
general all three terms are important. 

In order to use the normal equation the direction of the flow or the rate 
of change of speed must be known . Let /3 be the angle measured counter- 
clockwise about the zenith from the horizontal pressure force b# to the 
horizontal unit normal n as shown in fig. 7-200. ft is also the angle 



Section 7-20 



200 



between the isobar p and the flow direction s. Evidently the normal and 
tangential components of the pressure force are b n = b H cos ft and 
b a = bjj sin 0, respectively. Since v = b 8 , the normal pressure force may 
be expressed in terms of the horizontal pressure gradient and the rate of 
change of the speed as follows: 



At points where the current is gradient 0, 6 8 , and v are all zero; the 
current flows momentarily along the horizontal isobars with constant or 
extreme speed. In general the current overflows slightly across the 
isobars toward lower or higher pressure (see 7-15). 

As shown in fig. 7-206 (valid for the northern hemisphere) a given wind 
speed is compatible with any normal pressure force at a given latitude. 
In the diagram v nj b nj and c n are measured from the point /. Since <p 
and v are given, the horizontal Coriolis force is constant; c n extends to 
the right from / to the fixed terminal point G~, 




f 



.^ 



n- 



.-n 



FIG. 7-20a. 



FIG. 7-206. 



Let the normal pressure force extend from / to a variable terminal 
point B. If B is at / the flow is inertial, for b n = 0. If B is to the left 
of /, the flow is baric. And if B is to the right of /, the flow is antibaric. 

The horizontal centripetal acceleration v n is determined by the normal 
equation as the sum of b n and c n . The terminal point of v n then lies to 
the right of B by the length c n - ICT. 

When B is at G, the horizontal centripetal acceleration is zero; so 
KH = and the flow is geostrophic along the straight horizontal path g. 
If B is to the left of G, the flow is cyclonically curved to the left of g. 
And if B is to the right of G, the flow is anticyclonically curved to the 
right of g. 

When B is at /, the path is the inertial path i. When B lies to the left 
of / between G and /, the flow is anticyclonically curved between g and i. 
And when B lies to the right of /, the flow is anticyclonically curved to 
the right of i. 



201 



Section 7-21 



It is clear that cyclonic flow and geostrophic flow are always baric. 
Anticyclonic flow is baric when the path is less curved than the inertial 
path, and antibaric when the path is more curved than the inertial path. 

7-21. Maximum speed. We recognize in arbitrary horizontal flow 
many characteristics of zonal flow. Cyclonic flow and baric, inertial, 
and antibaric anticyclonic flow all appear in fig. 6- lie. Geostrophic 
flow also occurs, but only at the equator. 

In zonal flow we found that the horizontal pressure field builds up to a 
maximum and then decreases as the angular speed increases anticycloni- 
cally. In the northern hemisphere this maximum field occurs when the 
relative angular speed has the value 2. The pressure field in arbitrary 
anticyclonic flow behaves similarly, as illustrated for the northern hemi- 
sphere in fig. 7 -2 la. Here the latitude and the anticyclonic curvature 
are fixed. Hence c nj v n , and consequently b n are determined only by the 
speed. The speed increases linearly along the line labeled v. The 
terminal points of the abscissas c n , v n , and b n are plotted for all speeds 
between v = and v = v t -. The horizontal Coriolis force acts to the right 
and is a linear function of the speed. So the terminal point of c n de- 
scribes a straight line through 
v = 0. In anticyclonic flow the 
horizontal centripetal accelera- 
tion is directed to the right 
and is a quadratic function of 
the speed. So the terminal 
point of v n describes a parab- 
ola through v = 0. When the 
speed is zero, b n is zero; and 
b n is also zero when the speed 
is inertial. Between these two 
speeds the horizontal pressure 
field is baric; b n rises para- 
bolically to a maximum and 
then diminishes. The maximum occurs at one-half the inertial speed, 
as in zonal flow. 

This result may be established analytically. When the normal equa- 
tion is differentiated with respect to v for a fixed curvature and latitude, 
we find 



ov 



'n max 




Left: FIG. 7-21a. Right: FIG. 7-216. Maxi- 
mum normal pressure force. 



The normal pressure force is a maximum when 56 n /dv = 0. The speed 



Section 7-21 202 

at which this maximum occurs is denoted by Umax- From the above 
equation and 7*18(3) we have 

/IN Vi 

(1) *w - - ~ - 9 * 

2 



The angular speed of maximum flow is from 748(5) 

(fy\ _. TT * z , 

Wmax max "2 "" ~~ cos 0' 

This result may be checked for the case of zonal flow. The angular 
radius of curvature is then the colatitude, and (2) becomes <o max = - ft as 
required. The corresponding angular speed of the horizontal projection 
of the flow is from 748(4) 

(3) <o 2max = fl max KH = = Q z . 

For a given latitude and anticyclonic curvature the same normal 
pressure force (b n \ = b n2 ) occurs at v = v\ and at v = v 2 , as shown in 
fig. 7 '21&. That two speeds satisfy the normal equation is to be expected 
from the quadratic form of the equation. The diagram shows that v\ 
and v 2 are symmetric about t> max . Hence 

(4) vi + v 2 = 2z; max - Vi. 

Although both speeds v\, v 2 are dynamically possible, only the smaller 
speed occurs in large-scale anticyclonic currents. The reason for this 
will be given later in section 1145. There we shall examine the mecha- 
nism by which atmospheric circulation is generated. 

The speed of a large-scale anticyclonic current must satisfy the relation 

(5) v ^ <; max . 

Anticyclonic speeds greater than z; max will be called abnormal. Not only 
is the faster current in baric anticyclonic flow abnormal, but inertial 
currents and antibaric anticyclonic currents are also abnormal. There- 
fore antibaric flow and inertial flow seldom occur in the atmosphere. 
Even where an isobaric surface is level over an extended region, inertial 
flow as described in section 748 will not often develop; rather the entire 
region will have no wind. That is, when b n = 0, the solution of the nor- 
mal equation satisfying (5) is v = rather than the abnormal solution 

V = Vi. 

Equation (5) and its consequences apply only to large-scale anti- 
cyclonic flow in the atmosphere. All solutions of the normal equation 
are observed in small-scale mechanically produced vortices, in small-scale 



203 Section 7-22 

atmospheric eddies produced by friction, and possibly in other small-scale 
atmospheric vortices. 

722. Solution of the normal equation. The values of v g , Vi, and v c 
can be computed from the equations 



9O 

7-18(3) ^- 



7-19(2) 

where Km fl, and b n refer to arbitrary horizontal flow. These speed 
values may be regarded not as the actual current speeds but as speed 
parameters characterizing arbitrary flow. As parameters, v g , v^ and v c 
need not satisfy the conditions required for geostrophic, inertial, or 
cyclostrophic flow. For example, although v+ as a real current speed 
cannot be negative, Vi is negative as a parameter describing cyclonic flow. 
The three parameters are not independent. They are linked by the 
relation 

CD ?. fc fc 9 

In terms of any two of tftese parameters, the normal equation 
7-16(2) K H v 2 4- 212*1; - b n - 

may be simply expressed in three ways. Dividing the normal equation 
by K //, we obtain the first relation: 

(2) v 2 - V *,-v 2 c = Q. 

When (1) is substituted into (2), we find the second relation: 

(3) v 2 - Vi(v - v g ) - 0. 

The third relation is derived from (2) by division with v 2 , and subsequent 
substitution of (1): 

v 2 v 

(4) i| + ^~ 1 = - 

Since v g , v, and v c have the same sign for corresponding flow in both 
hemispheres, the equations (2, 3, 4) are free from the arbitrary sign con- 
ventions of 746(2). 



Section 7-22 



204 



Each of the expressions (2, 3, 4) contains the unknown v and two 
known parameters. Isopleths of v can then be drawn in a diagram with 
any two parameters as coordinates. Usually atmospheric flow is 
strongly influenced by the pressure field. Near the equator the flow is 
almost cyclostrophic; in higher latitudes the flow is almost geostrophic. 
Therefore the most convenient graph for representing all latitudes is 
constructed from (4). However, if we are mainly concerned with the 
higher latitudes (|^| ^ 30), (3) should be used. And in equatorial 
latitudes (M ^ 30) (2) should be used. 




FIG. 7-22a. 

In fig. 7-22a is shown the graph for (3), solving the normal equations 
in higher latitudes. The coordinates have been chosen so that the iso- 
pleths of v will appear as straight lines. For this purpose the linear 
coordinates must be v g and l/V{. 

The only curved line in the diagram is the envelope of the isopleths. 
Every isopleth is tangent to the envelope at a point halfway between the 
intersections of the isopleth with the two axes. Since the Vi scale is 
reciprocal, the Vi coordinate of this tangent point is Vi = 2v. From 
7-21(1) 2z; max ; so any point on the envelope represents maximum 
anticyclonic flow. It also divides the isopleth into two branches. The 
lower branch represents the flow v ^ z/ max , and the upper branch repre- 
sents the abnormal flow v ^ p max . 

The diagram is divided into four quadrants. In the lower quadrants 
the flow is cyclonic (v,- < 0), and in the upper quadrants the flow is anti- 
cyclonic (vi> 0). In the right quadrants the flow is baric (v g > 0), 



205 



Section 7-22 



and in the left quadrants the flow is antibaric (v g < 0). The positive 
horizontal axis represents geostrophic flow (v=*v g ), and the positive 
vertical axis represents inertial flow (v 




FIG. 7-226. 



Negative speeds have no physical significance and are not repre- 
sented by isopleths on the diagram. As a result two regions of the dia- 
gram are not covered by isopleths. These regions represent impossible 



Section 7-22 



206 



flow. The lower left quadrant represents antibaric cyclonic flow. This 
type of flow is impossible, for, as indicated in section 7-20, cyclonic flow 
must be baric. The region to the upper right bounded by the envelope 
represents impossible baric anticyclonic flow for which b n > 6 nmttx 
(see fig. 7-216). 

We shall give the complete practical solution of the normal equation 
in the form (3). The independent variables are p t T* 9 H n , <p, and 6. 
The required values for a current are v and D. 

For practical use the upper branches of the isopleths, representing 
abnormal anticyclonic flow, are deleted, so only the right-hand baric 



101=5 



a =784 



Anticyclonic 
v=15.7 
D = 6.1 




FIG. 7-22c. 

quadrants remain. The lower cyclonic quadrant may be superposed 
over the upper anticyclonic quadrant by folding along the horizontal 
v g axis. The cyclonic isopleths then slope upward to the right, and the 
anticyclonic isopleths slope upward to the left. The parameter v g is 
found by joining fig. 7-l7e to the bottom edge of the superposed quad- 
rants. The parameter Vi is found by turning fig. 7-18a over, cutting it in 
half along the coordinate <p 30, and joining the higher latitude half 
to the left-hand side of the superposed quadrants. Both joinings are 
permissible, for the v g isopleths intersect <p = 30 in a linear v g scale, and 
the Vi isopleths intersect <p = 30 in a reciprocal Vi scale. The resulting 
graph shown in fig. 7-226 gives v - v(p,T*,H n ,<p,6). An example of the 
procedure is illustrated in fig. 7-22c for the following data: p 100 cb, 
r*- 0C, H n = 3, ?- 45, 0= 5. The geostrophic speed and dis- 
placement are obtained as in fig. 7-17/. If the flow is cyclonic, then 
v - 9.7 m s"" 1 and D = 3.8. And if the flow is anticyclonic, then 
v- 15.7 m s" 1 and D - 6.1. 



207 Section 7-23 

In this example the cyclonic speed is less than the geostrophic value, 
and the anticyclonic speed is greater. The graph shows clearly that 
cyclonic flow is always subgeostrophic and that anticyclonic flow is 
always supergeostrophic. Analytically this result is obtained from (3), 
which shows that v 2 = Vi(v v g ) > 0. For cyclonic flow (vi < 0) we 
have v < v g , and for anticyclonic flow (vi > 0) we have v > v g . 

7-23. Horizontal curvature of the streamlines. The curvature KH 
is the horizontal curvature of the path of a particle. Since it is impos- 
sible to tag individual particles of air, the path cannot be observed 
directly. It is true that the approximate path may be inferred by suc- 
cessive displacements on consecutive weather maps. However, this 
method is both unsatisfactory and tedious. Fortunately the curvature 
of the path can be obtained from the streamline curvature. The hori- 
zontal streamlines are readily available on a synoptic constant-level 
weather map. They are curves everywhere tangent to the*wind direc- 
tion. The streamline and the path through a given point are both tan- 
gent to the velocity at that point. But in general the two curves have 
different curvatures. 

The relation between the curvature of the path and the curvature of 
the streamline at an arbitrary point P is obtained as follows: Let $H 
denote the angle, measured counterclockwise, from east to the projec- 
tion of the wind direction on the horizontal plane at P. At a given time 
\[/H is a function ^// = $H(SH)I of the arc length s// along the horizontal 
projection of the streamline. If the flow pattern is steady, this function 
is the same at all times. If the flow pattern is changing, the wind direc- 
tion at every point is also a function of time. The wind direction along 
the streamline through P is then 



Consider now the particle initially at P. During the time element dt 
this particle moves the infinitesimal distance ds = dsn along its path 
which is tangential to the streamline. The corresponding change of tyu 
on the particle is then 

t\\ Jt 

(i) ^ 

The horizontal curvature of the path is defined by the angular turn of 
the wind along the horizontal projection of the path: 

7-07(1) 

We shall denote the horizontal curvature of the streamline by 



Section 7-23 208 

This curvature is defined by the angular turn of the wind along the hori- 
zontal projection of the streamline at a fixed time : 

>./. __ 
(2) K HS 



SH 



The speed of the particle is v = ds/dt - ds H /dt. 

When (1 ) is divided by dsn and the above notations are introduced we 
find 



f^ K K j. 

(3) KH-KBS + - 



The expression d^^/d/ is the local turning of the wind at the point P. In 
practice the local turning of the wind is determined by the wind record- 
ings at a fixed station, or approximately by the wind change at the 
station between consecutive maps. 

Equation (3) gives the path curvature in terms of the streamline 
curvature and the local turning of the wind. The two curvatures at a 
given point in the atmosphere are equal only if .the wind direction at that 
point does not change. In particular, if the wind direction is independ- 
ent of time at every point in the atmosphere, that is, if the flow is 
steady (see section 7-15), then the path and streamline curvatures are 
everywhere equal. This result confirms the previous statement that for 
steady flow the streamlines and paths coincide. 

Since the atmosphere is actually characterized by changing flow 
patterns, the streamline and path curvatures are usually different. The 
two curvatures need not even have the same sign. We must then be 
careful about the usage of the words " cyclonic," " geostrophic," and 
" anticyclonic." Heretofore, we have characterized the flow of individ- 
ual particles. However, in synoptic practice the flow is designated as 
cyclonic, geostrophic, or anticyclonic according to the instantaneous 
streamline pattern. 

When (3) is substituted into the normal equation 7-16(2), we find 

(4) KH&? + (* + ~ ) - &n - 0. 

This relation may be expressed in the same form as the normal equation 
by letting the coefficient of the linear term be denoted by the symbol 
2Q zS . Thus 

dj,rr 

(5) 20,5-20, + ^- 

The normal equation then becomes 

(6) Kffsv 2 + 2to zS v -b n - 0. 



209 Section 7-23 

When the normal equation is expressed in the form (6), the coeffi- 
cients can be evaluated directly from meteorological data. Since the 
horizontal streamlines almost coincide with the horizontal isobars, 
KHS may usually be replaced by the horizontal curvature of the hori- 
zontal isobars. 

If the angular radius of curvature of the horizontal streamlines is 
denoted by 6s, the normal equation may be solved for v= 
v(p,T*,H n ,ft z s,Os)by the methods and graphs introduced in the previous 
sections. The geostrophic, inertial, and cyclostrophic wind parameters 
must, however, be replaced by the fictitious parameters v ff s, v^s, and 
v c s respectively. These new parameters are defined by the equations 
7-17(2), 7-18(3), 7-19(2) wherein U zS replaces Q z and Kns replaces^. 
In order to compute the values of v gS , Vis, and v c $ graphically the 
linear O z scale should be replaced by a linear Q z s scale, and & 2 s itself 
must be computed from (5). However, only the coordinate labels of the 
graphs are changed; the graphical operations are the same. 



CHAPTER EIGHT 
WIND VARIATION ALONG THE VERTICAL 

8-01. Geostrophic gradient flow. We have stated in the last chap- 
ter that a horizontal current above the surface layer flows nearly along 
the horizontal isobars. In general the current overflows across the hori- 
zontal isobars toward lower or higher pressure (see section 7-15). But 
this overflow is usually so slight that the current is nearly gradient. 

Moreover we have seen that cyclonic flow is subgeostrophic, and anti- 
cyclonic flow is supergeostrophic. Since a broad horizontal current 
above the surface layer is not often strongly curved, large-scale horizon- 
tal flow is approximately geostrophic, except in equatorial regions. 
Horizontal flow in the free atmosphere is then approximately geostrophic 
gradient flow. 

In order to examine the wind variation along the vertical we shall 
assume that the flow is both geostrophic and gradient. The velocity of 
this flow will be denoted by v g and will hereafter be called more briefly 
the geostrophic wind. Although only geostrophic flow will be investi- 
gated here, the deviation of this flow from any horizontal flow can be 
estimated qualitatively. If the actual wind velocity v is considered as 
the resultant of the velocity v g and a deviation velocity, the following 
analysis applies only to the geostrophic velocity. However, since v is 
often nearly equal to v g , the analysis of geostrophic flow yields several 
important approximative rules. 

The horizontal acceleration of a horizontal current is, from section 
7-13, 

VH = vt 4- K f jv 2 n.. 

When the current is both geostrophic (Kn = 0) and gradient (v = 0), 
the horizontal acceleration v// is zero. Therefore the equation 703(2) 
of relative motion in the horizontal plane becomes 

(1) - b// + c//. 

That is, the horizontal pressure force and the horizontal Coriolis force 
are in complete balance. 

Since by 7-12(1) the horizontal pressure force is aVnp, and since by 
741(7) the horizontal Coriolis force is 2Q Z xv, equation (1) may be 
expressed as 

(2) 2Q 2 xv a = b# = 

210 



211 Section 8-02 

The scalar form of this vector equation of geostrophic gradient flow is 
(3) 2Q,v g = ft*. 

Here bn occurs, rather than the b n of 7-17(2), for gradient flow is along 
the horizontal isobars. 

8*02. Isobaric slope. The speed of a geostrophic gradient current 
determines the isobaric slope. Let the acute angle between an isobaric 
surface and the horizontal level be 6 P . Then, as shown in fig. 8-02, the 
isobaric slope is 



(1) tan e p 



b z 5n 



Here dz p isf the rise of the isobaric surface above the horizontal level 
through the normal distance -8n. 

Since b z = g* and &// = 2tt z v gi we also have, from 7-14(1), 

(2) tan e p = -^ = ^ 

g g g 

The isobaric slope is then determined dynamically by the strength of 
the geostrophic wind. Even for the strongest speeds occurring in the 
atmosphere the isobaric slope is small. For example, when v g = 100 
m s"" 1 and v = 45 the isobaric slope is , 

about 1/1000. * 

From the hydrostatic equation the 
dynamic thickness of an isobaric layer 
is given by the mean specific volume 
of the layer. The variation along H- 

the vertical of the isobaric slope and 

,, r .1 , i - i FIG. 8-02. Isobaric slope, 

consequently of the geostrophic wind K 

depends on the variation of the specific volume within the isobaric 
layer. If the specific volume has no variation within the isobaric 
layer, the layer is barotropic, and the two isobaric surfaces bounding the 
layer have the same slope. So the geostrophic wind does not change 
throughout the layer. But if the specific volume varies within the iso- 
baric layer, the layer is baroclinic and is inflated in the direction of 
increasing specific volume. So the slope of the upper bounding isobaric 
surface differs from the slope of lower bounding isobaric surface. 
Accordingly the geostrophic wind varies along the vertical. 

t The symbols z p and Tp of this and the following section do not represent the 
same quantities as the'symbols z p and T p of section 4-30. 




Section 8-03 



212 



8 03. The thermal wind equation. The above argument may be 
developed analytically by differentiating the geostrophic wind equation 
along the vertical with respect to dynamic height. Since the latitude is 
constant along the vertical, application of d/d0 to 8-01 (2) gives 



(1) 2Q,x^~^. 

50 50 

Here 5v a /50 is the variation or shear of the geostrophic wind along the 
vertical. The wind shear may be clearly visualized, as shown in 
fig. 8-03a, by drawing the vectors v g for a given vertical at a given time, 
issuing from a common origin 0. The terminal curve of v g represents 
the distribution of wind along the vertical, with either dynamic or geo- 
metric height as the scalar variable. (See section 6-04.) This terminal 
curve will be called the shear hodograph. The wind shear is tangent to 
the shear hodograph and is directed toward increasing values of the 
height variable. 




FIG. 8-03o. Hodograph of wind dis- 
tribution along a vertical. 



FIG. 8-036. Definition of a p . 



(2) 



The variation of the horizontal pressure force along the vertical is 

j>/_ A ^ ^*7v,_ d 

,^ C 



b*""" ~b</> 
From the expressions 8-02(1, 2) for the isobaric slope we have 

(3) -^ 



a 00 



1 da bu 1 da fap 

~ ~ -- n = - n. 
a Oz g a OZ dn 



The partial differentiation symbols 5/d0 and V# are commutative. So, 
by the hydrostatic equation 5/>/d0 = -p, we have 



(4) 



213 Section 8-03 

When the horizontal volume gradient is expressed in the natural system 
and the equations (3, 4) are added, we find 



d<t> a ds a \dn dz dn 

Consider now the projection into the horizontal plane of the isosteric 
lines drawn on an isobaric surface, as shown in fig. 8'036. These lines 
define a horizontal field which will be denoted by a p . The ascendent 
V#<*p of this field will be called the horizontal isobaric volume ascendent. 
In the natural system this ascendent is expressed 

da p da p 



Since v g is along the horizontal isobars, the variations da p and da along 
the wind are equal. Therefore, 

da p da 
ds ds 

However, in the vertical plane normal to the wind direction the specific 
volume varies in both the n and z directions. That is, a = a(n,z). So 
an arbitrary variation da in the vertical (w,z)-plane is given by 

da ^ da 
da = dn + dz. 
dn dz 

In particular, when the variation is taken along the isobaric surface, 
da = da p and dz = dz p . Therefore, 

da p da da dz p 
dn dn dz dn 

The above expressions for the natural components of the horizontal 
isobaric volume ascendent show that (5) may be written 

(6) ^ = _I^E. 

d<t> a 

Consequently we have, from (1), 

(7) 2Q Z x = 



a 



This is the mathematical formulation of the dependence anticipated 
qualitatively in the last section. Evidently the geostrophic wind has no 
shear through a barotropic layer, since a p is constant for such a layer. 
The specific volume is not immediately given by aerological data, so 



Section 8-03 214 



for practical use (7) must be expressed in terms of the virtual tempera- 
ture. Since the pressure variation along an isobaric layer is zero, 
logarithmic differentiation of the equation of state gives 

(8) 



a i 

Therefore equation (7) for the geostrophic wind shear becomes 



This relation, known as the thermal wind equation, could also have been 
anticipated qualitatively. For within an isobaric layer an increase of 
virtual temperature always occurs with an increase of specific volume. 
Therefore the layer is inflated in the direction of increasing temperature. 
Usually the virtual temperature correction is so small, in comparison 
with the spatial variation of temperature, that T* may be replaced by T. 
Although the following argument applies strictly only to the virtual 
temperature, we shall for convenience write T for T*. 

8*04. Isothermal slope. The atmosphere is often analyzed synopti- 
cally by constructing the fields of the atmospheric variables in a series of 
constant-level charts. Although the horizontal temperature gradient is 
directly accessible from these charts, the isobaric horizontal temperature 
gradient, which occurs in the thermal wind equation, is not so easily 
available. We shall show in the next section that the difference between 
these two gradients is negligible. First, however, we shall derive another 
expression for the wind shear. 

Equations 8-03(6, 8) may be combined to give 

(1) ~ "TTT = " r~ ' 

00 a T 

Substituting here for 5b#/50, we have, from 8-03(2, 3, 4), 



- 
a a a 00 

To express this equation in terms of the temperature instead of the 
specific volume, differentiate the equation of state logarithmically. We 
then find the following vector and scalar relations: 

V//<* Vjyr _ Vnp Vgr bjy 

a " r" p " T pot ; 

!^_i^_!^_i<>r j_. 

a 50 " r &0 " p d0 " T 50 + pa 



215 Section 8-04 

If these relations are substituted in (2), the term bn/pa occurs twice 
with opposite signs. From (1) the logarithmic volume and tempera- 
ture ascendents are equal, so equation (2) becomes 



v//r p v//r 
~~'~ 



"" *' 



Comparison of equations (2, 3) shows that each symbol a in (2) has been 
replaced in (3) by the symbol T. 

The expression (dr/d<)b// in (3) is determined by the isobaric slope, 
for, by 8-02(2), we have 

ar dr&tf dr 

T- b// = - -- n = tan p n. 

O0 OZ g OZ 



Similarly the expression VnT in (3) is determined by the isothermal 
slope. Let OT be the acute angle between the isothermal surface and the 
level surface. Moreover, let n T be a horizontal unit vector, whose linear 
coordinate is UT, normal to the horizontal isotherms and pointing toward 
colder air. The isothermal slope then depends upon the rise 8z? of the 
trace in the (w^, 2) -plane of the isothermal surface above the horizontal 
level through the normal distance ^ buy. Thus 

8z T 
tan BT = ^ - 

OUT 

The upper sign (minus) is required when the temperature decreases with 
height. This is the usual case in the atmosphere. However, in layers 
where the temperature increases with height, the lower sign (plus) is 
required. 
The horizontal temperature ascendent is then given by 

br &r dz T ar 

- n r = - n r - - tan O T TL T , 
OUT 02 OUT Oz 



where all variations are taken in the (wy, 2) -plane. When the above 
expressions for the isobaric and isothermal slopes are substituted in (3), 
we have 

-jr* - J. -^ ( tan e T n T - tan p n). 
Therefore the thermal wind equation 8-03(9) takes the form 

>^ 1 PvT 1 

(4) 20, * f - - ~jr ^ tan 6>rnT ~ ten ^ : 

O<p JL OZ 

According to this formula the geostrophic shear may be attributed 



Section 8-04 216 

partly to the slope of the isobaric surface and partly to the slope of the 
isothermal surface. The isobaric slope is determined by the geostrophic 
speed and is always very small (see section 8-02). Since no similar 
dynamical control restricts the slope of the isothermal surface, any iso- 
thermal slope may occur in the atmosphere. 

8*05. The approximate thermal wind equation. We shall now show 
that the part of the shear attributed to the slope of the isobars is so small 
that it may be neglected for practical purposes. Suppose that the iso- 
thermal surfaces are horizontal. Accordingly OT = 0, and the shear may 
be attributed entirely to the isobaric slope. Division of the scalar form 
of 8-04 (4) by 2& z v g = g tan O p then gives 



Here the magnitude of the shear is proportional to the lapse rate. 
The maximum lapse rate ordinarily found in the atmosphere is the dry 
adiabatic, yd = l/c p d- For this lapse rate the percentual wind shear is 

1 |dv,| 1 



Vg 50 C pd T 



3.6% per dyn km, 



where the numerical percentage has been evaluated at T = 0C. Even 
this maximum strength of the shear is extremely small; in fact it is 
smaller than the error of measurement of the wind. Consequently the 
part of the shear attributed to the isobaric slope may be neglected for all 
practical purposes. Therefore, by 8-04(3), a good approximation to the 
thermal wind equation is obtained when the horizontal isobaric tempera- 
ture gradient is replaced by the horizontal temperature gradient. Thus 
the approximate form of equation 8-03(9) is 



Whenever the horizontal temperature gradient gives appreciable geo- 
strophic shear, the above approximate thermal wind equation may be 
used. Comparison of this equation with the equation of geostrophic 
flow 8-01 (2) shows that the wind shear and the horizontal temperature 
gradient have the same relative orientation as the wind and the horizon- 
tal pressure gradient. So the following law, similar to the baric wind 
law, holds for the wind shear : The geostrophic wind shear is directed along 
the horizontal isotherms, with low temperature to the left of the shear in the 
northern hemisphere, and with low temperature to the right of the shear in the 
southern hemisphere. 



217 Section 8-05 

This rule gives the direction of the geostrophic wind shear. For 
numerical computation of the strength of the shear (1) must be expressed 
in scalar form. In practical application the strength of the shear is 
measured by the magnitude |Av^| of the velocity variation through a 
finite layer of dynamic thickness A</> = gAz. Let the mean temperature 
through the layer be T, and let Aw^ be the distance between horizontal 
isotherms drawn for the interval AT". The scalar form of (1) is then 



(2) 



Az 



On upper level maps the horizontal isotherms are usually drawn for a 
temperature interval of 5C. If the distance between isotherms, 
measured toward lower temperature and expressed in degrees of latitude, 
is II Tj then Aw r = 1.11 x 10 5 f/ r and |Ar| = 5. The deviation of the 
mean temperature of a layer in the troposphere from the value T = 0C 
does not appreciably alter the value of the shear. Therefore equation 
(2) is approximately given by 

Az 



In weather reports the upper winds are given for every 1000 ft along 
the vertical. The shear at any one of these levels is measured by the 
shear of the wind from the level 1000 ft lower to the level 1000 ft higher. 
Thus Az = 2000 ft = 610 m. Moreover the speed is reported in miles per 
hour rather than in meters per second. Let &u g be the magnitude of the 
geostrophic shear, expressed in miles per hour. From the conversion 
table in section 1-06 the shear |AvJ is then given by Au / 2.237. With 
these values for |Av ff | and Az, the above equations may be written 

Z/VAw0sin <p 15.1. 

This form of the approximate thermal wind equation, first suggested by 
Neiburger, corresponds to equation 7-17(4) for the geostrophic wind. 

Comparison of the formulas for the geostrophic wind and for the geo- 
strophic wind shear show that the two equations are completely analo- 
gous. The geostrophic wind formula may be used to gauge the distance 
between horizontal isobars from wind reports. And the geostrophic 
wind shear formula may be used to gauge the distance between horizon- 
tal isotherms from wind shear reports. However, it should always be 
remembered that the practical use of these formulas is based on the 
assumption of geostrophic gradient flow. Although this type of flow 
is actually seldom realized, it often does give a fair approximation to the 
real flow in the free atmosphere above the friction layer. 



Section 8-06 218 

8-06. Analysis of the shear hodograph. The wind distribution 
along the vertical at a meteorological station is obtained directly from a 
pilot balloon observation. From the drift of the pilot balloon the wind 
is determined at equally spaced levels for instance, at every 1000 ft 
above sea level. The shear hodograph is constructed by marking the 
terminal points of the wind velocities drawn from the origin of a polar 
coordinate diagram. The consecutive terminal points are then con- 
nected by straight line segments as shown in fig. 8-06a. The real hodo- 
graph is, of course, a smooth curve. But it is not advisable to draw it 
with more details than the actual observations indicate, particularly if 
the hodograph has many irregularities and kinks. The terminal point 
of the surface wind is labeled 5, and the succeeding terminal points of the 
upper level winds are labeled by numbers indicating the elevation of the 
wind levels in thousands of feet above sea level. Although the wind 
vectors are drawn here for illustration, they are usually omitted. The 
mean shear through each 1000-ft layer is represented by the directed 
segment from the terminal point of the wind at the bottom of the layer 
to the terminal point of the wind at the top of the layer. 




10 mph 

FIG. 8-06a. 

Let fig. 806a represent the shear hodograph drawn from a pilot balloon 
observation taken in the northern hemisphere. Here the hodograph 
has not been drawn in the surface or friction layer S-> 3. For in the 
friction layer, usually about 3000 ft deep, the wind cannot be considered 
geostrophic. The wind distribution in the surface layer requires a 
separate analysis which will be presented in the next chapter. How- 
ever, at higher levels the reported wind and the shear may often be 
assumed to be geostrophic. By the approximate thermal wind equation 
this shear is along the horizontal isotherms, with colder air to the left. 
Thus, in the layer 3 8 of the diagram the mean horizontal isotherms 
run approximately northwest-southeast, with colder air to the northeast 



219 



Section 8-06 



of the station. And in the layer 8-> 13 the mean horizontal isotherms 
run approximately southwest-northeast, with colder air to the northwest. 
As pointed out by Rossby and collaborators, the shear hodograph 
shows qualitatively the horizontal direction toward the regions of maxi- 
mum or minimum vertical stability. The direction of the mean hori- 
zontal isotherms for the two layers 3-> 8 and 8 > 13 has been indicated 
by two intersecting straight lines in the simplified shear hodograph 
shown in fig. 8-06&. In the four sectors bounded by these lines the verti- 
cal temperature distribution is different. In the sector opening toward 
the east warm air lies over cold air, so the vertical temperature distribu- 



N 

W-4 E 
S 




Warm 
Cold 




Vo 

Warm /\ 

XWarm\ 
' Warm %% 

FIG. 8-06&. 



tion is relatively stable. In the opposite sector opening toward the 
west, cold air lies over warm air, so the vertical temperature distribution 
is relatively unstable. Therefore, in this example the direction toward 
the region of maximum vertical stability is toward the east, along the 
shear at the level 8. 

The following rule gives the direction toward the region of maximum 
or minimum vertical stability at a level where the shear hodograph is 
appreciably curved: The shear points toward the region of maximum 
stability if the concave side of the hodograph encloses colder air, and points 
toward the region of minimum stability if the concave side encloses warmer 
air. This rule, valid in both hemispheres, must be qualified further. At 
the level to which the rule applies the hodograph must be significantly 
curved as well as appreciably curved. That is, the level must lie between 
two fairly deep layers through which the hodograph is nearly straight, 
as in the example. The above qualified rule may be derived mathe- 
matically by differentiation of the thermal wind equation along the 
vertical (d/d<). 

Finally, the shear hodograph also shows qualitatively the tendency, or 
local change with time, of the stability along the vertical. In the lower 



Section 8*06 220 

layer 3 -* 8 of the shear hodograph (fig. 8-06a) the wind blows across the 
isotherms from a warmer region, so the temperature in that layer will 
rise. At the level 8 the wind blows along the isotherms, so the tempera- 
ture of that level will not change. And in the upper layer 8-* 13 the 
wind blows across the isotherms from a colder region, so the temperature 
in that layer will fall. The consequent change of the vertical tempera- 
ture distribution is illustrated in fig. 8-06. The full-drawn sounding 
curve represents the temperature distribution along the vertical at the 
time of observation. As explained above, advection of temperature will 
change this temperature distribution toward the dashed sounding curve. 
Consequently the sounding will become less stable. 

The following rule applies to the tendency of the vertical stability at a 
level where the shear is along the wind : The sounding becomes less stable 
if the concave side of the hodograph encloses colder air, and becomes more 
stable if the concave side encloses warmer air. This rule, valid in both 
hemispheres, applies only if the shear hodograph is both significantly and 
appreciably curved. It may be derived mathematically by local time 
differentiation of the thermal wind equation (d/d)- 

Both the above rules are useful in weather analysis and forecasting. 
By the first rule the distribution of vertical stability in space can be 
inferred from the shear hodograph. And by the second rule the local 
change of vertical stability with time can be inferred. However, these 
rules should be applied only where the shear hodograph is appreciably 
and significantly curved, and where the flow is approximately geostrpphic 
gradient flow. 

8-07. Fronts. We have examined the wind shear through an atmos- 
pheric layer in which the mass field, represented either by the specific 

Transitional 
layer x 




FIG. 8-07a. FIG. 8-076. 

volume or the density, varies continuously. We shall now examine the 
shear through a surface of discontinuity in the mass field. Such a sur- 
face is called a frontal surface or more briefly a front. 

A real front in the atmosphere is never a sharp discontinuity. As 
shown in fig. 8-07a, it is a transitional layer with rapid but continuous 
variation of the specific volume or of the density. However, it is often 



221 Section 8-08 

convenient to treat this transitional layer as a surface of discontinuity 
separating two air masses. At any point of the front the density under- 
goes an abrupt finite change from the lighter to the denser air mass. As 
shown in 8'07& the isosteric surfaces enter the frontal surface from one 
side, follow the front for a certain distance, and leave the frontal surface 
on the other side. 

8-08. The dynamic boundary condition. Consider any point P on 
the frontal surface of discontinuity. As P is approached from within the 
dense air mass, the physical variables (p, a, p, T, v) approach definite 
values, which will be denoted by unprimed symbols. And as the pointP 
is approached from within the light air mass, the variables also approach 
definite but in general different values, which will be denoted by primed 
symbols. The difference of these two values, the dense air mass value 
minus the light air mass value, will be indicated by the symbol A. Thus 
the density difference at the front separating two air masses is 

(1) p-p'=Ap>0. 

This condition can be taken as the definition of the front. In the special 
case where Ap = 0, the air mass discontinuity disappears, and the front 
is nonexistent. 

Although the mass field is discontinuous at the front, dynamic princi- 
ples require that the pressure field be continuous. That is, the pressure 
value at a given point on the front must be the same in both air masses. 
For otherwise the pressure gradient and pressure force would be infinite. 
Therefore, at any point in the frontal surface we have 

(2) A/> - 0. 

This condition is known as the dynamic boundary condition. 

The differentiation symbol 6 has been used to denote variation at a 
fixed time between neighboring points of space. The variation of the 
pressure difference through a front between two neighboring points on the 
frontal surface is, from (2), 



(3) *(*p) = A(#) = 0. 

This equation expresses the dynamic boundary condition in differential 
form. It is valid only when the differentiation is taken in the frontal 
surface. 

Let the differentiation be performed along the vector line element 5r F 
in the frontal surface. Then from 443(1) we obtain the following 
expression for the dynamic boundary condition : 

(4) A(S) . A(W/0 - a>fA(Vp) - 0. 



Section 8-08 



222 



This equation shows that the frontal surface is perpendicular ^to the 
variation of the pressure gradient through the front. Let the variation 
of the pressure gradient be denoted by the vector F. Thus 



(S) 



F= A(-V/0 



The dynamic boundary condition (4) requires that F be a vector per- 
pendicular to the frontal surface, as illustrated ity fig. 8'08a. 




FIG. 8-08a. F = A(- Vp). 



FIG. 8-086. F = - 



The same result can be obtained differently, as illustrated in fig. 8-086. 
First assume that the cold air mass extends beyond the front and that 
the pressure throughout the entire frontal region is given by the field p. 
Next assume that the warm air mass extends beyond the front and that 
the pressure throughout the entire frontal region is given by the field p'. 
The extended pressure fields p, p' are represented in the diagram by 
dashed isobars dividing the frontal region into unit isobaric layers. 
Consider now the fictitious situation where both fields p, p f are extended 
through the front. The two sets of isobaric units layers intersect each 
other and divide the frontal region into tubes of parallelogrammatic 
cross section. It is readily seen from the diagram that the unit surfaces 
of constant A/> are the diagonal surfaces of these tubes. But according 
to (2) one of these surfaces, namely, A = 0, is the frontal surface. 
Therefore the frontal surface must be normal to the vector F = -V(A), 
as shown by equation (S). 

8*09. Application of the dynamic boundary condition. The general 
relation between the frontal surface and the pressure field is given by the 
dynamic boundary condition. Some of the consequences of this con- 
dition have already been explained. Further consequences will now be 
derived which may be applied directly to the weather map. 

The difference between the two values of a physical quantity at a front 
has been indicated by the symbol A. The arithmetic mean of these two 



223 



Section 8-09 



values will be indicated by a superior bar (""). For instance, the mean 
value of the density at a front is p = (p + p')/2. 

Since p = p f at a front, the equation of state p = p/RT* shows that the 
density and temperature differences through a front are related by 

* P -/ r*'-r* AI 



Hence, from 8O8(1) the virtual temperature difference AT 1 * is always 
negative. In the following we shall distinguish the air masses at the 
front by virtual temperature, rather than by density; we shall call the 
dense air mass cold and the light air mass warm. Henceforth we shall 
write T for T* as in the thermal wind equations. 

The front may be stable or unstable, depending upon the arrangement 
of the cold and warm air masses. The front is stable if the cold air flows 
in a wedge under the front and warm air flows above. And the front is 
unstable if the warm air flows in a wedge under the front and cold air 
flows above, for in this case thermal convection would immediately de- 
stroy the frontal discontinuity. There- 
fore we shall in the following consider 
only stable fronts. 

The slope of the frontal surface may 
be obtained by the method used for the 
isobaric and isothermal slopes. Let F 
be the acute angle between a frontal 
surface and a level surface. As shown 




FJG 8 . 09a Fronta , slope> 



in fig. 8*09a, the slope of a stable front with the cold air flowing below 
the front and the warm air flowing above is 



(2) 



tan 0, = 



The horizontal and vertical components of F are obtained from the 
definition: F= A(V/>). Thus the vertical component F z is 



F, = F-k--A(V/>r 



- (I) 



Since for all practical purposes the hydrostatic equation 'bp/'bz = gp 
is valid, we have finally 

(3) F z 



In order to find the horizontal component of F, we shall introduce the 
horizontal orthogonal unit vectors t/?, n F whose linear coordinates are 
s F , n F respectively. The two unit vectors will be oriented so that t r 



Section 8-09 



224 



is along the front with n F to the left and pointing toward the warm air. 
For a stable front n F is along the horizontal projection of F. Thus 



(4) F H = F-n, = -A(Vp-n,) = -A ( ^- 
so the slope of a stable front is given by 

(5) tan 6 F = - 



The inequality (4) gives one condition that must be satisfied by the 
horizontal pressure field near a stable front. Since the vector F is 
normal to the front, a second condition may be obtained as follows: 



(6) 



F-t, 



The two conditions (4): A(d/dj,) < and (6): A (&/>/&$,,) = 0, 
derived from the dynamic boundary condition, restrict the variation of 
the horizontal pressure field in the neighborhood of a front. Nine possi- 
ble combinations of the two conditions are shown in fig. 8-09. These 
diagrams show the front and the isobars in a level surface. The cold air 
covers the upper half of each diagram and the warm air covers the lower 
half. 



dS* 



dp' 






H 



H 



H 



H 






FIG. 8-096. Nine possible horizontal pressure fields in the neighborhood of a 

front. 



225 Section 8-10 

In the diagrams of the center column the horizontal pressure gradient 
has no component along the front, so the horizontal isobars are parallel 
to the front. In the diagrams of the left-hand column the horizontal 
pressure gradient has a component to the right along the front. And 
in the diagrams of the right-hand column the horizontal pressure gradi- 
ent has a component to the left along the front. Notice that the hori- 
zontal isobars intersecting the front have a kink pointing from low to high 
pressure. This rule, valid in both hemispheres for any stable front, is 
extremely useful in the frontal analysis of constant-level weather maps. 

8-10. The kinematic boundary condition. The dynamic boundary 
condition is a restriction on the pressure field at the front. There is also 
a restriction"on the field of motion at the front, for the two air masses on 
either side of the front cannot move so that a void develops between them 
or so that they interpenetrate. That is, the velocity components VN, 
v' N of the cold and warm air masses normal to the front are equal. Thus, 

(1) 



This condition is known as the kinematic boundary condition. 

Let N F be the unit vector along the vector F, normal to the frontal sur- 
face. The velocity components VN, v r N are respectively v*N F , v^N^. 
So the kinematic boundary condition may be written 

A(vN F ) = !VAv = 0. 

Multiplication of this equation by the magnitude |F| gives the following 
useful form of the kinematic boundary condition : 

(2) A(vF) = F-Av = 0. 

This equation requires that the velocity difference between the cold and 
warm air masses be along the front. That is, the two air masses may 
slide along the front with any tangential velocity difference. 

The kinematic boundary condition is illustrated in fig. 8-10. Here the 
plane of the page represents any plane intersecting the front, and the 
arrows represent the projection of the velocities v, v 7 into this plane. 
The arrangement of the nine diagrams corresponds to the arrangement of 
the nine diagrams in fig. 8'096. 

Since the front separates the two air masses, the speed of the front is 
given by the velocity component normal to the front. Therefore in the 
diagrams of the center column the front is stationary (UN = 0), and in the 
diagrams of the right- and left-hand columns the front is moving 
(VN T* 0). A moving front actively pushed by the cold air mass (VN > 0) 
is called a cold front. And a moving front actively pushed by the warm 
air mass (VN < 0) is called a warm front. Consequently, the diagrams of 



Section 8- 10 



226 



the left-hand column represent cold fronts, and the diagrams of the 
right-hand column represent warm fronts. 

If the flow in the two air masses at the front is horizontal, v z = 0. So, 
by expansion of the scalar product vF, the kinematic boundary condi- 
tion (2) may be written 

(3) 



HereF/f is a horizontal vector normal to the front in the horizontal level. 
Moreover, if the front is stable F# points toward the warm air mass. 
Therefore the unit vector n^ is along F#. 



v N >Q 



V s0 



v N <0 



\\\ 



\ \ \ 



\ \ \ 



\\\ 



FIG. 8-10. Nine possible velocity fields in the neighborhood of a front in the 

northern hemisphere. 

8-11. Front separating two arbitrary currents. The dynamic and 
kinematic dynamic boundary conditions apply to any front. Since well- 
defined moving fronts are always marked by extensive cloud, systems, 
atmospheric flow near moving fronts cannot be horizontal. The vector 
F normal to the front is defined as the difference of the pressure gradient 
through the front. When the equation of motion is expressed in the 
absolute frame, we have from 6-09(2) 

(1) -F- A(V) = A[ P (g -v a )]. 



And when the equation of motion is expressed in the relative frame, we 
have from 6-19(3) 

(2) -F 



227 Section 8-12 

The frontal slope may be obtained by the method used in section 809 
as the ratio between the horizontal and vertical components of F. When 
the front separates two arbitrary currents the general formula for the 
frontal slope is too complicated for practical evaluation. However, the 
frontal slope is readily evaluated when the flow in the two air masses 
separated by the front is zonal or geostrophic. A front separating two 
constant zonal currents will be called a zonal front. A zonal front is a 
stationary surface of revolution about the axis of the earth, and is com- 
pletely defined by its trace in a meridional plane. A front separating two 
geostrophic currents is called a geostrophic front. In section 8-13 we shall 
show that a geostrophic front also is stationary. 

8'12. The zonal front. In zonal flow the absolute acceleration is, 
from section 641, V = -wR. Therefore equation 8-11(1) becomes 

(1) -F- A[p(g+2R)]. 

Only small percentual variations of p and o> will be considered here, so 
we may treat the symbol A as a differentiation symbol. Therefore, since 
g a and R do not vary through the front, (1) may be written 

(2) -F - A P (g + w*R) + 2pw Aw a R. 
To simplify this equation we shall define 

gA = go + WR, 

/3 = 2poJ a Aco a . 

The vector g^ will be called the apparent gravity, for to an observer mov- 
ing zonally with the absolute angular speed c3 the force of apparent 
gravity replaces the force of gravity g = g a + 12 2 R. Moreover, a sur- 
face <j>A normal to gA, called an apparent level, replaces the geopotential 
level <t> normal to g. Introducing the quantities g^ and ft into (2), we 
have 

(3) -F~gAAp+/3R. 

Since ft has the same sign as Aw a , we may distinguish the following 
three cases according to the sign of Aw a . 

(i) Aca a < 0. Here the warm air rotates faster than the cold air. 
The orientation of -F is shown in the upper diagram of fig. 842a, and 
the corresponding orientation of the stable zonal front is shown in the 
lower diagram. The cold air is then on the polar side of the front and 
the warm air is on the equatorial side. This is the usual geographical 
distribution of temperature. 

(ii) Aw a = 0. Here the angular speed does not vary through the 
front. The isobaric surfaces in both air masses and the frontal surface, 



Section 8- 12 



228 



being normal to the apparent gravity, coincide with the apparent levels. 
The sea surface may be considered as a front separating the dense mass 
of water from the light mass of air. When the ocean and atmosphere 
have the same zonal rotation, the sea surface is an apparent level. 
And if, in particular, w a = 12, the sea surface is a geopotential level as 
indicated earlier. 

(iii) Ao? a > 0. Here the cold air rotates faster than the warm air. 
The orientation of -F is shown in the upper diagram of fig. 8*126, and 
the corresponding orientation of the stable zonal front is shown in the 
lower diagram. The cold air is then on the equatorial side of the front 





<0 





FIG. 8-12a. 



FIG. 8-126. 



and the warm air is on the polar side of the front. Although this is an 
unusual geographical distribution of temperature, such a front may 
develop in winter on the polar side of a cold continental region, with open 
water off the coast. 

In all figures the apparent level has been shaded. If o5 = 12 this 
apparent level may be taken as the surface of the earth. If Z5 a < 12 the 
apparent level slopes downward toward the equator. And if o> > 12 
the apparent level slopes downward toward the poles. 

Notice that in both cases (i) and (iii) the stable zonal front has the 
faster rotating air on the equatorial side of the front and the slower rotat- 
ing air on the polar side. Therefore the horizontal shear of the two zonal 
currents at the front is always cyclonic. 

We shall now find the slope of the frontal surface with reference to the 
apparent level. Let FAH and FA Z be the apparent horizontal and verti- 
cal components of the vector F. The acute angle between the front and 
the apparent level will be denoted by 6 AF . Therefore, in analogy to 



229 Section 8-13 

8-09(2), the apparent frontal slope is given by 

/ F 
tan 6 AF = 



Let the angle between the apparent level and the axis of the earth as 
shown in fig. 8'12c be denoted by <PA, called the apparent latitude. The 
apparent horizontal and vertical components of R are then R sin <PA and 
R cos <PA respectively. Consequently we have for the apparent horizon- 
tal and vertical components of F, from (3) : 



FA. - gA AP - PR cos M . 
The frontal slope is then given by 




- 
tan 9 AF tan *> A |0|U sin ^ FlG - 8 ' 12 *- Apparent latitude. 



Here the plus sign should be taken when Aco a < 0, and the minus sign 
when Aw a > 0. Usually the frontal angle 0^.is small, compared with the 
apparent latitude <PA, so the reciprocal of tan <?A may be dropped from 
the above equation. Thus 



2co a sin (p a _ 

(4) tan OAF = ------ : --- P- 

gA Ap 

For practical use the density difference should be expressed by the 
temperature difference according to 8-09(1). Moreover, for all practical 
purposes the apparent level coincides with the geopotential level. So 
the final approximate form for the slope of a zonal front is 



The two equations (4, 5) are always nearly equivalent. In particular, 
when w a = 12, they are exactly equivalent. 

8'13. The geostrophic front. When the vectors in equation 8*11(2) 
are projected into the horizontal plane, we obtain 

-Ftf - A(V///>) - A[p(c/7 - V /7 )]. 

For geostrophic gradient flow, v// = and c# = -2ti z v g n (see section 
? 1 1 ) . Moreover, 12 Z does not change through the front. Consequently 
the above equation may be written 

(1) F# = 2 



Section 8- 13 230 

Let us form the scalar triple product F//'F# xk. The volume of the 
parallelepiped having the three vectors F//, F//, k as edges is, of course, 
zero. Accordingly from (1) we have 

0= 2 



The factor multiplying p inside the parentheses may be simplified as 
follows: 

xk = 00FVt 



By the kinematic boundary condition this factor has the same value on 
either side of the front, and consequently may be taken outside the 
symbol A. Thus we have finally 

(2) 2^F 7/ - V(7 Ap - 0. 

The density difference Ap is by definition positive, so the scalar product 
F#*V0 is necessarily zero except at the equator. Therefore the geo- 
strophic flow is parallel to the front. Since the front moves only if the 
air masses have a velocity component normal to the front, geostrophic flow 
in the two air masses separated by a front is possible only if the front is 
stationary. 

The slope of a stationary front separating two geostrophic currents 
may now be expressed by the geostrophic speed. From 8-09(2) the 
frontal slope is equal to the ratio Fn/F z . The value of the vertical 
component F z is, from 8-09(3), F z = gAp. And since the unit vector TL F 
is along F/f for a stable front, the horizontal component F H is, from (1), 

(3) F H - F/7-n^ - 2n,A(p^n-n F ). 

Since the unit vector n F is 90 to the left of t,,., and the unit vector n 
is 90 to the left of t, the angle between n and n^ is also the angle between 
t and t F . Therefore 

VglL'TLp = Vgt'tp = Vg'tf. 

And the frontal slope is 



The slope of a stable front is always positive, so A(pv <7 ) - t f . is positive 
in the northern hemisphere and negative in the southern hemisphere. 
Consider a stable geostrophic front in the northern hemisphere. The 
horizontal unit vector t F is along the front, with warm air to the left and 
cold air to the right. Therefore, the vector difference of geostrophic 
momentum A(pv^) also must have warm air to the left and cold air to the 
right, as shown in the three diagrams of fig. 8-13a. 

These diagrams show that the horizontal shear of the geostrophic momen- 



231 



Section 8- 13 



turn at a stable geostrophic front is always cyclonic. We have already seen 
that a similar rule applies to a stable zonal front. Although the rule 
has been derived only for stationary fronts, observations show that it 
usually applies to moving fronts. For this reason the diagrams of 
fig. 8 10 have been drawn with cyclonic shear. 



P'V' 



FIG. 8-13a. Shear of momentum at a geostrophic front. 

We have shown in section 7-16 that the direction to the left of the flow 
in the southern hemisphere corresponds to the direction to the right of 
the flow in the southern hemisphere. Therefore, the direction of the 
arrows showing the geostrophic momentum in fig. 8-13a should be re- 
versed in the southern hemisphere. However, the rule stated above is 
still valid. 

A useful form for calculating the frontal slope is obtained by making 
the approximation 



g 



Equation (4) then becomes 
(5) tan 



The magnitude of the first term on the right is, from 8-02(2), the slope 
tan 9 P of an isobaric surface in a geostrophic current with the speed v g . 
Consequently the magnitude of the second term on the right is, from 
8-09(1), given by 



Here the plus sign should be taken when the warm air has the larger 
speed, and the minus sign when the cold air has the larger speed. 

The expression on the right of equation (6) is tabulated below for 
T = 0C and for reasonable wind and temperature differences through a 
front at 45 latitude. These values indicate that the isobaric slope 
( < 1/1000) can be neglected for rough approximations. So the practi- 
cal formula for the slope of a geostrophic front is 



(7) 



tan d F - 
g 



f. 



Section 8-13 



232 



This formula, known as Margules' formula, is equivalent to the approxi j 
mate expression 8*12(5) for the slope of a zonal front. Moreover, as we 
shall now show, Margules 1 formula also expresses the slope of a transi- 
tional frontal layer. 



|A7l 


lAv.l 


10 m s" 1 


20 m s" 1 


30 m s" 1 


5C 


1/175 


1/88 


1/58 


10C 


1/351 


1/175 


1/117 



Real fronts in the atmosphere are transitional layers with rapid but 
continuous variation of the temperature and wind through the layer. 
If the flow throughout the transitional layer is geostrophic, the slope of a 
stationary transitional layer may be derived from the practical thermal 
wind equation. 

We shall assume that the horizontal isotherms are parallel to the fron- 
tal layer. This assumption is not too restrictive, for the horizontal iso- 

steres and isobars are almost parallel to a station- 
ary front. The coordinate UT is then measured 
normal to the horizontal trace of the frontal 
layer. Therefore, as shown in fig. 8-13&, the 
2\z F slope of a stable transitional layer is given by 
'Op ^ 

FIG. 8-13&. tan e * = ~ ~^ F = A~ * 

When the geostrophic shear Av /Az F is measured vertically through 
the frontal layer and the temperature variation AT is measured hori- 
zontally through the layer, the approximate thermal wind equa- 
tion 8-05(2) may be written in the form 



Transitional 

layer 
An* 




2GL 



tan 6 F = ~ = 7- 



]Ay,| - 



This equation is Margules' formula for a transitional frontal layer. The 
wind variation through a continuous frontal layer has the same approxi- 
mate form as the wind variation through a frontal surface of 
discontinuity. 



CHAPTER NINE 

WIND VARIATION ALONG THE VERTICAL IN THE 
SURFACE LAYER 

0*01. Dynamics of friction. The principles which have been derived 
in the two foregoing chapters are not applicable in the lowest layer of the 
atmosphere. The motion is here strongly influenced by friction and 
must be studied with an equation of motion in which the frictional force 
is included. The aim of this chapter is to determine how far up from the 
ground this frictional influence extends, and to find the main characteris- 
tics of the motion in the layers below, where friction operates. 

Let the frictional force per unit mass be denoted by m. The equation 
of motion in the layer of frictional influence next to the ground is then 

(1) v=b+c + g + m. 

The frictional force depends in a rather complicated way upon the state 
.of the motion and the physical state of the atmosphere. The general 
problem of the effect of friction upon an arbitrary motion is not yet 
accessible to rigorous dynamical treatment. In what follows we shall, 
therefore, confine our discussion to the simple case of steady great circle 
flow. 

9 '02. Steady great circle flow. The state of motion to be considered 
in the surface layer is great circle flow, constant throughout each level, 
and having the proper variation in direction and magnitude from level to 
level. The problem will be to determine what the velocity variation 
with height must be if this flow shall remain steady under the influence of 
friction. 

Since the motion is constant great circle flow, the horizontal accelera- 
tion is zero, and the horizontal equation of motion becomes 

(1) = b// + c// + m//. 

The balance of these three forces is shown schematically in fig. 902a. 
The horizontal Coriolis force is expressed in terms of the velocity as 
follows : 

7-11(7) c*- -2Q,xv. 

The motion may be resolved into one component equal to the geostrophic 



Section 9-02 



234 



velocity along the isobar, and an additional component which we shall 
call the geostrophic deviation, denoted by u (see fig. 9*02a). Thus 

(2) v - v a + u. 

Substituting this in the expression for c#, we have 

This equation gives the actual Coriolis force as the resultant of two 
vectors, one being the Coriolis force in a motion with the geostrophic 
wind, and the other the Coriolis force resulting from the deviation from 
the geostrophic wind. By definition the Coriolis force of the geostrophic 
wind is balanced by the pressure force: 

8-01(2) b//= 2Q 2 XV0. 

Substituting this expression and (3) in (1), we obtain for the frictional 
force 

(4) m// =* 20,; xu. 

This equation states that the deviation u of the actual wind from the geo- 
strophic wind must be such that the Coriolis force resulting from the 
deviation will balance the frictional force. The three equations for 





P 




FIG. 9-02a. 



FIG. 9-026. 



state: (i) that the force triangle, fig. 9-02a, is similar to the 
velocity triangle, with the factor of proportionality 2Q* (taken equal to 
one in the diagram) ; (ii) that the force triangle is turned 90 to the right 
of the velocity triangle in the northern hemisphere. 

Equation (4) is the basic equation for steady motion in the surface 
layer. When the proper expression for ma is known, the geostrophic 



235 Section 9-03 

deviation is obtained by integration, and the total motion is subse- 
quently obtained from equation (2). 

The first attempt to study the influence of friction between the atmos- 
phere and the surface of the earth was made by Guldberg and Mohn in 
1875. They assumed that the frictional force is directed opposite to the 
velocity and that its magnitude is proportional to the speed. The 
velocity triangle then becomes a right triangle, as shown in fig. 9-026. 
If the factor of proportionality k is assumed to decrease with height, the 
angle \l/ which the wind makes with the isobars has a maximum at the 
ground and decreases with elevation. 

This solution explains in a qualitative way some of the observed 
features of the motion in the surface layer, notably that the wind has a 
component toward lower pressure. However, although it is true that 
the frictional force in a qualitative sense may be expected to act against 
the motion and tend to slow it down, there is no reason to believe that the 
expression for the frictional force is so simple as Guldberg and Mohn 
assumed. In order to derive a rational theory for atmospheric friction 
it becomes necessary to examine the physical mechanism of fluid resist- 
ance in some detail. 

9-03. The viscous stress. Consider a layer of fluid enclosed between 
two rigid horizontal plates separated by a distance z, as in fig. 9-03a. 
The lower plate is at rest, and the upper plate is kept in steady horizontal 
motion with the velocity v. When steady conditions are established, 
experiments show that the velocity of the fluid increases linearly from 
zero at the resting plate to the velocity v at the moving plate; in other 
words the shear is constant throughout the layer. This motion develops 
as a consequence of the viscosity or internal friction which arises from the 
irregular random motion of the fluid molecules. 




4 




T 
R 


A 




T U 



FIG. 9-03a. FIG. 9-036. 

Experiments show further that in order to keep the upper plate in 
steady motion it is necessary to apply a tangential force which is propor- 
tional to the velocity of the plate and inversely proportional to the dis- 
tance z between the two plates. If this force, referred to unit area of the 
plate, is denoted by T, we have 

(1) T~- 



Section 9-03 236 

As no accelerations exist, the force on the upper plate is balanced by an 
equal and opposite force on the lower plate in order to keep it at rest, as 
shown in fig. 9-036. Consider now a vertical column with unit cross 
section, extending from the lower plate to an arbitrary horizontal plane 
AB between the two plates. The force -r on the lower face of this 
column has to be balanced by a force T on the top face of the column in 
the plane AB. This force, which is caused by the frictional interaction 
between the fluid layers on both sides of the plane, is called the viscous 
stress. In the ideal experiment described here we find that the viscous 
stress has the same value at every point in the fluid layer, and from (1) 
it is proportional to the constant shear in the layer. 

Since the stress is caused by molecular action, it is evident that it can 
depend only on the velocity distribution in the immediate vicinity of the 
plane across which it acts. We may therefore generalize the above result 
to the case of horizontal motion which is constant on each horizontal 
plane, but where the velocity has an arbitrary variation along the verti- 
cal, as in fig. 8-03a. Also here we may expect that the stress at any level 
is proportional to the shear, dv/dz, at that level. Denoting the factor 
of proportionality by M we have then 

dv 

(2) T-M-. 

This relation, which was discovered by Newton, is called Newton's for- 
mula for the stress. The proportionality factor /* is called the viscosity. 
The viscosity is found to be a physical property of the fluid. It is inde- 
pendent of the velocity distribution, the dimensions of the system, and 
so forth. 

9-04. The viscosity of a perfect gas. Since the stress is created by 
the molecular motion, we should be able to derive Newton's formula 
theoretically if the molecular motion is known. In a liquid fluid nothing 
is known of the molecular motion, but for a gas Newton's formula can be 
derived from the kinetic theory, as shown by Maxwell. 

The kinetic theory states that the molecules of a gas move with a 
random distribution of velocities, and that the kinetic energy of this 
motion is the internal heat energy of the gas. Being random oscilla- 
tion, such motion contributes nothing towards moving the gas as a 
whole. Let the molecules in addition to this random heat motion 
have an ordered streaming motion which is constant in planes parallel 
to AB in fig. 9'04. Consider the conditions at the plane AB. As a 
result of the random heat motion, molecules from below will pass up- 
ward, and from above downward, each carrying along with them the 



237 Section 9-04 

streaming momentum corresponding to the layer where they experi- 
enced their last impact. There is accordingly a transport of momen- 
tum through the plane, whereby the faster moving upper layer loses 
momentum, and the slower moving lower layer gains momentum. 
The process may be likened to the case of two trains moving in the same 
direction along parallel tracks, but ^ v 

with somewhat different speeds. If I V v + z ** 

the passengers jump from one train to ^ / 

the other, the faster train will be slowed ^ j - y v - B 
down, and the slower train will be L / 

accelerated. According to Newton's ' - *>' v -~ L 

second law the rate of change of mo- 

1 4. 4 C FlG - 9 ' 04 - 

mentum is equivalent to a force ex- 

erted by the upper layer on the lower layer. This force is the viscous 

stress. 

We shall now express this idea in mathematical form. Let c be the 
average speed of the heat motion when the flow velocity is subtracted. 
Let m be the mass of each molecule, and let N be the number of mole- 
cules per unit volume. To simplify matters, assume that one-third of 
the molecules move perpendicular to the plane, one-half of this number 
N/6 moving downward, and the other half upward. The number pass- 
ing downward per second through unit area is then Nc/6. These mole- 
cules experienced on an average their last collision at the distance of the 
mean free path L, and each has the streaming momentum m[v -f (5v/dz)L] 
characteristic of this level. Thus per unit area and time at the plane AB 
we have : 

the downward momentum transport => ^Nmc I v -f L 1 
And by the same reasoning we have: 

the upward momentum transport = %Nmc ( v - L J 

The product Nm is the mass per unit volume, or the density p.^. 'The net 
downward transport of momentum per unit area and time, 



is the stress which the fluid above the plane exerts on the fluid below the 
plane. This equation is Newton's formula, 9-03(2), and gives for the 
viscosity the expression 

(2) M - $pcL. 



Section 9-04 238 

This relation shows that the viscosity is a physical property of the fluid. 
Further, since pL is constant for a given gas, the viscosity is independent 
of the density and hence of the pressure of the gas. Since the tempera- 
ture is proportional to the square of the mean heat speed c, the viscosity 
is proportional to the square root of the temperature. The independence 
of pressure is well substantiated by experiments, but the increase with 
temperature is somewhat greater than this theory indicates. 

9-05. Viscosity of air and water. Viscosity is measured by letting 
the fluid flow through a circular pipe under the influence of a known 
pressure gradient and recording the rate of discharge. Some values of 
the viscosity of air and water, obtained from such measurements, are 
given in table 9-05. The dimensions of viscosity are, from 9-04(2), 
[/*] = [ML" 1 !""" 1 ], and the numerical values in the table are in mts units. 

TABLE 9-05 
VISCOSITY IN MTS UNITS 

TC Air Water 

1.7 X 10~ 8 1.8 X l<r 6 

100 2.2 X 10~ 8 0.2 X 10~ 6 

In water the viscosity decreases rapidly with increasing temperature, as 
is generally true in all liquid fluids. In air the viscosity increases with 
temperature, in qualitative agreement with Maxwell's theoretical expres- 
sion 9-04(2). However, it is seen that the actual increase is more rapid 
than the theory predicts. For both air and water the viscosity is prac- 
tically unaffected by pressure changes, even up to 100 atmospheres. 

The viscosity of air and water is quite small. For comparison the 
viscosity of glycerin at 20C is about 9 X 10~ 4 mts units. 

9 '06. The f fictional force. In the special case of horizontal motion 
with shear the frictional force is readily expressed by the shearing stress. 
Consider a vertical column of unit cross section extending from the level z 
to the level z 4- dz. See fig. 9-06. Let -r be the stress exerted on the 
bottom face by the fluid below, and T + (dr/dz)dz the stress exerted on 
the top face of the column by the fluid above. These stresses may have 
different horizontal directions. The resultant of these stresses, 
(dT/dz)6z, is the frictional force on the volume element 8V = 8z. The 
frictional force per unit volume is therefore dr/dz, and the frictional force 
per unit mass is 

(1) m H =a 



239 



Section 9-07 



Note the analogy between this expression for the frictional force and the 
expression -aVp for the pressure force. Like the pressure, the stress is 
defined as a force per unit area. The pressure gradient is the pressure 
force per unit volume, and similarly the variation of the stress per unit of 
height is the frictional force per unit volume. To obtain the correspond- 
ing forces per unit mass the volume forces are multiplied by the specific 
volume. 

Only in the special case of horizontal motion with uniform velocity at 
each level has the frictional force the simple form (1). If the motion 
also has horizontal shear, 
tangential stresses on the 
vertical side faces of the 
fluid particles will result, 
and the horizontal gradients 
of these lateral stresses add 
to the frictional force. If 
the stress on a horizontal 
face is r 2 , and the stresses 
on the vertical faces normal 




FIG. 9-06. Variation of the stress with height. 



to the x axis and the y axis are respectively r x and T V , it can be shown 
that the general expression for the frictional force is 



(2) 



m 



dj + ? 



The analogy with the expression for the pressure force is here complete. 
In all large-scale atmospheric currents the horizontal shear of the wind 
is extremely small in comparison with the vertical shear in the surface 
layer. The lateral stresses T X and i y are then correspondingly small, and 
the frictional force is in the first approximation given by the simple 
formula (1). 

9 '07. Total mass transport in the surface layer. When the expres- 
sion 906(1) for the horizontal frictional force is introduced in the basic 
dynamic equation for steady motion in the surface layer, 9-02(4), we 
have 



(1) 





a - = 2\L Z x u. 

Oz 



When this equation is multiplied by the density, we have 

fr" ^ 

= 2Q 2 xpu. 



Integrating this equation along a vertical extending from the ground, 



Section 9-07 240 

where the stress is T O , to the top of the atmosphere, where the stress is 
zero, we have 

oo 

(2) -TO = 2Q 2 x / P u8z. 

o 

The integral on the right represents the total transport of mass resulting 
from the geostrophic deviation through a vertical column of unit cross 
section extending from the ground to the top of the atmosphere. Denot- 
ing this mass transport by F tt , 

00 

(3) F u - / puSz, 



equation (2) can be written 

(4) -TO- 2Q 2 xF tt . 

The physical interpretation of this equation is simple. Consider the 
total vertical column of air, having unit cross section and extending from 
the ground to the top of the atmosphere. No acceleration exists within 
this column, and the resultant of all the acting forces is therefore zero. 
The horizontal pressure forces acting on the column are balanced by the 
Coriolis forces which result from the geostrophic part of the motion. 
The only remaining external force is the stress -T O exerted by the 
surface of the earth on the bottom face of the air column. To balance 
this stress the air in the column must have a motion in addition to the 
geostrophic wind (that is, a geostrophic deviation) such that the sum of 
the resulting Coriolis forces throughout the complete column is equal 
and opposite to the stress. The total mass transport of the geostrophic 
deviation is therefore directed at right angles to the surface stress and is 
proportional to the stress, in accordance with equation (4). 

We may similarly define the mass transport F of the actual wind and 
the mass transport F^ of the geostrophic wind: 



(5) 



00 00 

'- / pv8z, F0= / 



When the equation v = v g + u is multiplied by p and the three terms are 
integrated from the bottom to the top of the atmosphere we have then 

(6) F = F, + F tt . 

Equations (4) and (6) are illustrated by fig. 907. 




241 Section 9-08 

A problem of considerable practical importance is that of evaluating 
the total transport of air across the isobars. According to (6) only the 
part F M of the total mass transport contributes to the cross isobar trans- 
port. The component of F M normal to the isobar can be evaluated from 
(4) when the direction and the magnitude of the surface stress T O are 
known. 

The surface stress is proportional to the wind shear immediately above 
the ground and is therefore directed along the wind at a short distance 
above the ground, say at the anemometer level where the surface wind is 
measured. It is rather plausible, 
and also borne out by theory 
(see H. U. Sverdrup: " Oceanog- 
raphy for Meteorologists," Pren- 
tice-Hall, 1942, p. 119) that the PX***^ j? * P 

magnitude of the surface stress at *^ *o 

any given station depends only FIG. 9-07. Mass transports in a vertical 

upon the anemometer wind speed. column from the ground to the top of the 

The cross isobar transport can atmosphere. 

therefore be determined entirely from an observation of the surface 

wind when the direction of the isobar is known. 

It should be noted that the above relation between the surface stress 
and the cross isobar transport does not involve any assumptions as to 
the wind distribution along the vertical. 

9-08. Wind distribution in the surface layer. We shall now turn to 
the main problem of this chapter, namely to determine the wind dis- 
tribution along the vertical in a steady straight current. To simplify 
the mathematical problem it will be assumed that the horizontal pressure 
force and, hence, the geostrophic wind have the same direction and mag- 
nitude at all levels, and further that the specific volume is independent 
of height. The latter assumption is not so serious a violation of actual 
conditions as it may seem, for it will be shown that the effect of friction 
is mainly confined to the lowest kilometer. Finally it will be assumed 
that the viscosity is independent of height. The justification of that 
assumption will be considered below. 

When Newton's formula 9-03(2) for the stress is substituted in 9-07(1), 
we have 



(1) OtfJL r-~2 = 2Q Z XU. 

In this equation the height 2 is the only independent variable. Differen- 
tiation with respect to height will in the remaining part of this section 
be denoted by primes. Since the geostrophic wind is assumed inde- 



Section 9-08 242 

pendent of height, the variation with height of the actual wind is equal 
to the variation of the geostrophic deviation. Equation (1) can then be 
written 

(2) ex/m" = 2Q 2 kxu. 

The three scalar parameters 12 2 , a, JJL are, according to our assumptions, 
constant along the vertical. They are replaced by a single constant /3 2 , 
given by 

(3) ? = -*-; 

an 

here ft 2 is positive in the northern hemisphere and negative in the 
southern hemisphere. Then equation (2) takes the form 

(4) u"=2/3 2 kxu. 

The solution of this homogeneous linear differential equation can be 
written down directly if the two-dimensional vector u is treated as a 
complex variable. A more elementary solution is given here. We 
introduce a natural system of reference t, n, k referred to the velocity u, 
such that 

(5) u = wt. 

The derivatives of the two unit vectors t and n are 

t' = 0'n, 

<6> 

where 0' = d0/dz represents the rate at. which the vector u turns with 
increasing height. Differentiation of (5) gives for the shear of u 

(7) u'-tt't+fl'wn. 

The shear is here expressed as the sum of its natural components tangen- 
tial and normal to the vector u. There is a close formal analogy between 
this development for the shear and the development in section 605 for 
the acceleration, the only difference between the two being that one 
represents differentiation with respect to height and the other differentia- 
tion with respect to time. When (7) is differentiated once more, and the 
relations (6) are considered, we have 

u" - \u" - 0' 2 u]t + [0V + (0'tt)']n. 
The vector product in (4) becomes 

kxu= z*(kxt) = wn. 



243 Section 9-08 

Substituting these expressions in (4), we obtain the two component 
equations 

(8) u" = 0'V 

(9) 0V + (0'w)' = 20 2 . 

The general solution of this simultaneous system (8, 9) is rather compli- 
cated. However, we are only interested in the special solution for which 
u remains bounded as z - oo. This is the boundary condition for the 
system (8, 9). If it were violated, we should have infinitely high veloci- 
ties at the top of the atmosphere, which is physically impossible. 

To find the special solution, we first assume that 0' is constant. Then 

(9) takes the form 

(10) 0V - f3?u. 
Differentiation of (10) gives 

(11) 0V'=/3V. 
Elimination of u between (8) and (10) gives 

(12) 0V'-^V. 

P 

Comparison of (11) and (12) shows that 0' 4 = /3 4 , whence for the system 
(8, 10) to be consistent we must have /2 = /3 2 . Since /2 is positive by 
physical definition, the plus sign applies to the northern hemisphere and 
the minus sign applies to the southern hemisphere. Restricting the 
further discussion to the northern hemisphere, we see then that (10) 
takes the form 

(13) u = Q'u. 

It is clear that (8) is a consequence of (13), so that the solution of the 
system (8, 13), i.e., the special solution of (8, 9) for constant 0', is obtained 
by solving (13) alone. 
The solution of (13) is well known to be 

(14) u = u e", 

where UQ is the value of u at z = 0. Now, in order that u be bounded as 
z > oo, we see from (14) that 0' must be negative. Since 0' 2 = j3 2 , it 
follows that 

(15) 0' = -ft 

where j3 is the positive root of ft 2 . We then have from (14) that 

(16) u = HOC-'*. 



Section 9*08 244 

Equations (15) and (16) give the northern hemisphere solution of the 
system (8, 13). It can be shown by more advanced methods without 
any assumption about 6 f that (15, 16) is the only solution of the system 
(8, 9) that obeys the boundary condition at the top of the atmosphere. 
In interpreting (16), the simplest condition would be that the velocity 
v at the ground be zero, whence from 9-02(2) w = |v a |. However, the 
dynamical conditions at the ground are more complicated than assumed 
here and must be studied with an equation different from (4). The 
solution of (4) can therefore not be extended to the ground, but only to a 
somewhat higher level (say 10 m above the ground). This level may 
arbitrarily be taken to be the " anemometer level " where the surface 
wind is measured. Let V be the velocity at the anemometer level, and 
let U be the corresponding geostrophic deviation (see fig. 9'08) ; thus 

v = V0 + U . 

Hence, when the height z is measured from the anemometer level, the 
UQ of (16) is equal to |u | = |v - v^|. The equation (16) then says that 
the magnitude of the geostrophic deviation decreases exponentially with 




FIG. 9-08. Ekman spiral. 

increasing height from the maximum value UQ at the anemometer level. 
The turning of the wind with height is given by (15). Here 6 is the 
angle from an arbitrary fixed direction to the direction of u. To simplify 
the formulas we shall measure the angle 6 from the direction of U at the 
anemometer level, so that = 0. The integral of (15) is then 

(17) e fa. 

The explicit solution at the anemometer level and above is given by 
combining equations (16) and (17); thus 

(18) u = u Q e e and 6 = -fa. 

The geostrophic deviation turns linearly with height in a negative sense, 
and its magnitude decreases exponentially with the angle of turning. 
The equation (18) is the equation for the hodograph in polar coordinates. 
It is the well-known expression for the logarithmic spiral. The solution 
is known as the Ekman spiral after W. F. Ekman, who solved the corre- 
sponding problem for the surface layer of the ocean in 1902. 



245 Section 9-09 

Fig, 9-08 illustrates schematically the velocity distribution in the 
layer next to the ground. It shows the velocity V and the geostrophic 
deviation UQ at the anemometer level, and also the hodograph of u. 
The latter is, of course, also the hodograph of the total velocity v. Two 
values of u, for the angles 6 = -^TT and = - ^TT, which correspond to 
the levels z = far/ 8 and z = |-7r//3, are indicated in the diagram. 

9 09. Relation between surf ace velocity and geostrophic velocity* The 

analysis in the preceding section does not give any information regarding 
the direction and magnitude of the surface wind (at the anemometer 
level). 

The behavior of the surface wind is determined by the dynamics of the 
bottom layer, from the anemometer level down to the ground. The 
theory of this layer will not be treated in this book. It must suffice to 
mention a few of the results. It can be shown that the turning of the 
wind in the bottom layer is negligible, and so the wind and the shear of 
the wind have here approximately the same direction. Applying this at 
the anemometer level, we have 

(1) V =KVo. 

The scalar factor of proportionality K depends upon the height of the 
anemometer level and the roughness of the ground. For example, if 
the anemometer level is at 10 m and the ground is open grassland, the 
proportionality factor is 55 m. 

A well-known geometrical property of the logarithmic spiral is that 
the tangent at any point makes the constant angle of 45 with the 
radius vector. This is readily shown from the formula 9-08(7) for the 
shear. Substituting here for u and 0' from 9-08(13, 15) we find 

(2) v' = u'= -0tt(t+n). 

The shear is directed along the tangent of the hodograph. From (2) 
the shear is parallel to the vector t -f- n, which makes an angle of 45 
with t and hence with the radius vector u. Applying this result at the 
anemometer level, where from (1) the velocity and the shear are parallel, 
we find that the angle between the directions of v and -UQ is 45, as 
shown in fig. 9*09. It follows then immediately from the diagram that 
Vgsin fa v e cos fa VQ, or 

(3) PO = v g (cos ^ ~ sin ^ )- 

Another relation between the surface wind and the geostrophic wind 
is obtained as follows. The magnitude of the shear at the anemometer 



Section 9-09 246 

level is, from (2), 

(4) K| 

Taking the magnitude in equation (1) and substituting from (4), we find 

(5) v Q 




v f 



FIG. 9-09. 

Further, by inspection of fig. 9-09, 

(6) v g sin \!/Q = o sin 45 

And finally when UQ is eliminated from (5) and (6) 

(7) VQ - 2KJ3v g sin ^ - 

This relation, together with the relation (3), determines the direction and 
magnitude of the surface wind. When the ratios between the two equa- 
tions are taken, both VQ and v g are eliminated and we find 

(8) cot ^o=l + 2/c/3. 

The angle ^ between the surface wind and the isobar is thus less than 
45, which also is directly apparent from the diagram. The actual 
numerical value of the angle, and hence of the surface wind speed, depends 
upon the magnitude of the two parameters K and ft. 

910. The geostrophic wind level. The velocity distribution in 
fig. 9-08 agrees at least qualitatively with actual observations in a region 
with straight isobars over level ground. The surface wind is consider- 
ably smaller than the geostrophic wind, and it has a component across 
the isobars toward lower pressure. With increasing elevation the wind 
speed increases, and its direction approaches that of the geostrophic 
wind. The lowest level where the wind becomes parallel to the isobars 
is called the geostrophic wind level. As shown in fig. 940, this level is 
reached when the /geostrophic deviation has turned an angle 
QH ** (f * + ^o) The height // of the geostrophic wind level measured 
from the anemometer level is determined by QH " pH When the 



247 Section 9-11 

above value for 0# is used, we find 

(1) H- J(iir + lM 

In the second expression to the right the value of has been introduced 
from 9-08(3). At the equator the height of the geostrophic wind level 
would become infinite. The physical reason for this is the fact that the 




FIG. 9-10. 

horizontal Coriolis force is zero at the equator, and hence no deflection 
of the motion can create forces which will balance the pressure force and 
the frictional force. The above discussion is therefore invalid close to 
the equator. 

The height II is proportional to V /* and hence increases with the 
viscosity of the air, which might be expected a priori. If the 
friction is caused by molecular action only, p, is the molecular 
viscosity. As an example take the 0C value, /z = 1.7 X 10~ 8 mts 
units, and a = 850 m 3 t"" 1 . These values introduced in (1) give for the 
height of the geostrophic wind level at 40 latitude the value 1.6 m. 
Under the influence of molecular viscosity the layer of frictional influ- 
ence in the atmosphere would only have a thickness of 1 to 2 m. 

We know by experience that the geostrophic level is found much 
higher up. The height varies widely, depending on the nature of the 
surface of the earth and the stability of the air. A rough average value 
at 40 latitude is about 1500 m, or roughly one thousand times the value 
derived above. Substituted in (1), this value of the height would 
require that the viscosity be about 1.6 X 10~~ 2 mts units, or roughly one 
million times larger than the molecular value. It is evident from this 
that the internal friction is created by an enormously more powerful 
agent than the molecular motion. This agent is the atmospheric 
turbulence, and the viscosity which is found in the example above is the 
turbulent viscosity or eddy viscosity n e . 

9- 11. The eddy viscosity. To complete the present discussion some 
qualitative remarks concerning the eddy viscosity will be made. It 



Section 9-11 248 

appears that the main characteristics of turbulent flow can be described 
by considering the motion as the resultant of a certain mean flow and a 
random eddy flow, the latter giving no net transport of mass in any 
direction. This is evidently analogous to the molecular heat motion 
superimposed on the streaming flow of the air (section 904). 

Consider the simple case where the mean flow is horizontal and is also 
steady and uniform in each horizontal level. The turbulent eddies will 
carry parcels of fluid from level to level, mixing rapidly into their new 
surroundings. The net result of this is a transport of mean momentum 
across the planes of constant mean flow. The slower streaming fluid on 
one side of the plane gains momentum at the expense of the faster stream- 
ing fluid on the other side. This momentum transport is equivalent to 
a tangential stress, called the eddy stress, across the planes of constant 
flow. It is equal to the rate of flow of momentum across unit area. 

To obtain an expression for the eddy stress it is assumed that the 
parcels of air which are affected by the eddies move a certain average dis- 
tance I before they lose their identity and mix with their new environments. 
The distance / is called the mixing length. It corresponds to the mean 
free path of molecular motion, although its physical definition is less 
precise. Let w be the average component of the eddy velocity perpen- 
dicular to the plane of constant flow. By reasoning similar to that in 
section 9-04 we find that the rate of momentum flow across unit area, 
and hence the eddy stress, is proportional to pwlfivfoz), where v is the 
velocity in the mean flow. The factor of proportionality is usually 
included in the definition of the mixing length, so 

dv 

(1) r=pwl 

oz 

This formula for the eddy stress is analogous to Newton's formula for 
the molecular stress. It brings out the formal analogy between the 
dynamical action of molecular and eddy motion. The expression 
pwl is dimensionally a viscosity and plays the role of viscosity in the 
above formula. It is therefore called the eddy viscosity and denoted 
by He'- 

(2) ^ - PWI. 

There is some similarity between this expression and Maxwell's formula 
9-04(2) for the molecular viscosity. But, contrary to the molecular 
viscosity, the eddy viscosity is not a physical property. The mixing 
length depends upon the roughness of the ground, the distance from the 
ground, and also on the velocity distribution in the mean flow. The 



249 Section 9-11 

vertical eddy component w depends upon the mixing length and the 
shear of the mean flow. 

Several methods exist whereby the eddy viscosity and its distribution 
with height can be determined directly from the vertical wind distribu- 
tion. From such determinations it is found that the eddy viscosity first 
increases from the ground upward, and then decreases again higher up. 
The actual values are found to vary within a wide range. Over a rela- 
tively level ground and with moderate stability the range is roughly from 
0.5 X 10~ 2 to 5 X 10~ 2 mts units, with an average value of about 10~ 2 . 
From the formula (2) this average value would correspond to a mixing 
length of 8 m and a vertical eddy velocity of 1 m s" 1 . 

The analysis of the frictional layer in the preceding sections is valid 
for a turbulent layer when the eddy viscosity is used, and when it is 
further assumed that the eddy viscosity is constant in the layer. See 
9-08(1). The above remarks would indicate that the assumption of a 
constant eddy viscosity is unjustified. Nevertheless, the solution for a 
constant eddy viscosity having the average value of the layer is a useful 
first approximation to what actually happens. 



CHAPTER TEN 
MECHANISM OF PRESSURE CHANGES 

10*01. Equation of continuity. One important physical principle 
which has not yet been considered is that of the conservation of mass. 
This principle states that no fluid mass can be created or destroyed. 
Consider an arbitrary fixed volume bounded by a fictitious closed bound- 
ary at any place in an air current. Air will then flow through this 




FIG. 10-01. Net inflow of mass along x axis. 

volume, passing in through the boundary from one side and escaping 
through the boundary on the other side. The principle of the conserva- 
tion of mass requires that the net inflow of mass into the fixed volume in a 
given time equal the increase of mass within the volume during the same time. 
To express this statement in mathematical form, let the fixed volume 
be an infinitesimal parallelepiped BV = 5x5ydz, as in fig. 10-01. The net 
inflow along the x axis into the volume 5 V during the time element dt is 



(pv x )8ydzdt - 



4- 






= - ( P v x )dVdt. 



Here pv x is a mean value for the area dydz. Similar expressions are 
obtained for the net inflow along the y axis and the z axis. The sum of 
these three expressions represents the total net inflow of mass into the 
volume 5 V in the time dt : 



-I"- 

Ldx 



d 

' + r- 



dVdt. 



This inflow of mass must cause an increase of density in the volume ele- 
ment from the value p at the time / to the value p -f (dp/dt)dt at the time 

250 



251 Section 10-02 

/ -f dt. The increase of mass in the volume 8V during the time dt is there- 
fore 



The principle of the conservation of mass requires that the two expres- 
sions (1) and (2) be the same. When the factors 8V and dt are divided 
out, we obtain the two equivalent expressions for the rate at which mass 
flows into a fixed unit volume. When these are equated we have 

I- 

This equation is known as the equation of continuity. The expression 
on the right can be written more conveniently in vector notation when 
the following convention is adopted. The operation symbol V used in 
442 may be considered as a vector with the rectangular components 
d/djc, d/cty, 5/dz. The part of equation (3) enclosed in brackets is then 
the scalar product of the vector operator V and the momentum vector 
pv; thus 

(4) V-(pv) = (pv x ) + (pvy) + (pv z ). 

Ox oy Oz 

The expression in (3) represents the rate of inflow or convergence of mass 
into unit volume. The quantity in (4), having the opposite sign, repre- 
sents the outflow and is called the divergence of the vector pv. It is 
sometimes denoted div(pv). Since pv represents the flow of mass the 
divergence V(pv) of this vector is called the mass divergence. 
With the notation (4) the equation of continuity takes the form 

dp 

(5) = -V(pv). 

dt 

An alternative form of the equation of continuity will be derived in the 
next section. 

10*02. Divergence. The compact vector expression for the mass 
divergence, V(pv), can be developed as follows: 

(1) V(pv) = v-Vp + pV'v. 

This formula is verified by performing the differentiation on the right in 
10'01(4) and expressing the result in vector notation. Accordingly the 
divergence operator V follows the rules of differentiation when operat- 
ing on the product of a vector and a scalar. 
The first term on the right in (1) is the scalar product of the velocity 



Section 10-02 252 

and the density ascendent. The second term on the right contains the 
expression 

, . &Vx bv y bv z 

(2) V-v= -+-* + -*, 

ox oy oz 

which is the divergence of the velocity field. The physical meaning of 
the velocity divergence is obtained as follows: Consider a moving ele- 
ment of air which at the time / fills the rectangular space 8V - dx8ydz of 
fig. lO'Ol. At the time t + dt this element will form an oblique parallele- 
piped whose angles differ infinitesimally from right angles. Its edges, 
to the first order in dt, are 



The volume of the element at the time / + d t is given, to the first order in 
ft, by the product of the three edges: 



- I" 1 

L 

When we divide by dV = dxdydz, we have 



1 

dt 



, 

5 F dt 

This equation shows that the divergence of the velocity measures the 
ate of expansion per unit volume of the moving air elements. Equa- 
:ion (3) is a purely kinematic relation which is a consequence of the 
geometry of the motion. 

If dM = pdV is the mass of the moving element, we have on account of 
,he conservation of mass that 

pdV = const. 
Jpon logarithmic differentiation, we obtain 



v 

p at 8V dt 

substituting here for the divergence from (3), we have 

dp 

; 4) - = -pvv. 

This is the second form of the equation of continuity. 

10*03. Horizontal divergence. Since the motion of the atmosphere 
s mainly horizontal, it is sometimes convenient to write the divergence 



253 Section 10-04 

as the sum of the horizontal divergence, 

(1) 



and the vertical divergence, &v z /bz. Thus according to 10-02(2) 



(2) 

dz 

The physical significance of the horizontal divergence is similar to that 
of the three-dimensional divergence. Consider, at the time /, the small 
horizontal area 5A = dxdy which forms the base of the parallelepiped of 
fig. 10-01. Even though the motion may have a vertical component we 
shall assume that this area element moves in the horizontal plane with 
the horizontal velocity components. When the argument of the last 
section is repeated for this area we find 

(3) 1 <">-** 

The horizontal divergence is then the rate of areal expansion per hori- 
zontal unit area of a fictitious element moving with the horizontal com- 
ponents of the motion. If the motion is strictly horizontal, the hori- 
zontal divergence is represented by the rate of areal expansion of the 
real fluid element per unit area. 

10*04. Individual and local change. The change at a fixed point of a 
physical variable is called the local change of that variable. The local 
changes of pressure and temperature, for example, are those which are 
recorded by a barograph and a thermograph at a fixed station. The 
local rate of change is denoted by partial differentiation with respect to 
time, b/d/. The change which occurs on a given particle during its 
motion is called the individual change. The individual rate of change is 
denoted by the differentiation symbol d/dt, or by the dot symbol. 

It is important to note the difference between the individual and the 
local rate of change. To find the relation between them, consider any 
one of the physical variables, for instance the density. In accordance 
with 4-04(3) the density field is analytically expressed as a function of 
the rectangular coordinates x, y, z, and the time t: p = p(x,y,z,t). The 
variation of the density from the point (x,y,z) at the time t to an arbi- 
trary neighboring point (x + dx, y + dy, z + dz) at the time / + dt is 
given by 

(1) dp = ^dx + ^dy + ^dz+^dt = drV P + ~-dt. 

d# by dz at at 



Section 10-04 254 

This expression has general validity for arbitrary variations dr and dt. 
We shall, however, specialize it so that dr is the displacement during the 
time dt of the particle which has the position (x,y,z) at the time /. 
dr/dt = v is then the velocity of the particle, and the corresponding dp 
is the individual change; thus 

dp dp 
(2) = + vVp. 

The individual rate of change is here expressed as the sum of the local 
rate of change and a second term, vVp, which is called the advective rate 
of change. The latter represents the change of density of the moving 
particle due to its motion into regions of different density. The for- 
mula (2) is readily verified for the following two special cases: (i) Equi- 
librium : The velocity is zero and therefore the advective change is zero. 
Every particle remains at rest and its individual change is identical to 
the local change in the field, (ii) Steady state: The field of density 
remains fixed in space and thus no local changes occur. The individual 
change on a particle can then only arise from a motion into a region of 
different density, i.e., from an advective change, in accordance with the 
formula. The relation between individual and local change has here 
been derived for the density. Obviously the derivation holds for any 
other physical variables in the atmosphere, vectors as well as scalars. 

The equation of continuity offers a good illustration of the local and the 
individual change of density. In its first form, 10-01(5), it expresses 
the local change of density as the mass convergence into unit volume. 
In its second form, 10-02(4), it expresses the individual change of 
density in terms of the velocity convergence. That the two forms are 
equivalent follows from the fact that the one is transformed into the 
other when the relation (2) between the individual and local change is 
used. Substituting, for example, from (2) into the second form, 
10-02(4), we have 

dp 

-pV-v 



dt 
or, after rearrangement, 

~ - (vVp + pV'v) = -V'(pv). 

This is the first form of the equation of continuity, 10-01 (5). 

10*05. Relative change in a moving pressure field. The pressure 
field in a constant-level map moves over the map in a more or less regu- 
lar way, and the internal change in shape and structure of the pattern is 



255 Section 10-06 

usually slow and gradual. Certain features of the pressure field for 
example, troughs, wedges, centers of high and low pressure can be 
identified from one map to another. The first step in the preparation 
of a weather forecast is to determine the displacement of the pressure 
field during the time of the forecast period. The second step is to evalu- 
ate or estimate the changes which will occur in the meteorological vari- 
ables relative to the moving pressure field during the same time. 

These relative changes are related to the local changes by a formula 
similar to 10-04(2). To find this formula consider an identifiable point 
in the moving pressure pattern, for example, the point of intersection of 
a wedge line with one of the latitude circles. Let this identifiable point 
move through the displacement dp during the time interval dt* From 
10-04(1) the relative change of density at this point during its displace- 
ment is 

dp 
(1) dip-dirVp + ^dt. 

The velocity of propagation of the pressure system is d$/dt = c, and 
therefore 

dip dp 



The relative change in the moving pressure field is thus the sum of the 
local change and the advective change due to the movement of the 
pressure field. 

The formula (2) holds for the change of any physical variable. Of 
particular interest is the relative change of the pressure itself: 

(3, *-*+** 

The relative change of pressure, dip/dt, which indicates the deepening or 
filling of the pressure pattern, is thus expressed as the sum of the local 
pressure change and the convective change caused by the movement of 
the pressure pattern. The relation between the relative and the local 
change was first studied by Petterssen (1933). From the formula (3) 
Petterssen derived a number of kinematic formulas for the movement of 
troughs, wedges, and pressure centers. A comprehensive discussion of 
these formulas and of their application to weather forecasting is found in 
Petterssen's book " Weather Analysis and Forecasting." 

10*06. The pressure tendency. In synoptic meteorology the local 
pressure change in the atmosphere is called the pressure tendency. The 
systematic study of the mechanism of local pressure changes was started 



Section 10-06 256 

by J. Bjerknes in 1937. One important application of Bjerknes' theory 
is that it explains the pressure changes occurring during the development 
of the cyclone. This cyclone theory will be found in sections 10-1 7-10-20 
below. First the tools of analysis must be developed, and some pre- 
liminary studies of simplified atmospheric models must be carried out. 
The treatment is essentially the same as that presented in a paper by 
J. Bjerknes and J. Holmboe.* 

10-07. The tendency equation. The new tool of analysis introduced 
by Bjerknes in 1937 was the so-called tendency equation. This equa- 
tion is obtained by a combination of the hydrostatic equation and the 
equation of continuity. 

The hydrostatic equation is assumed to be valid in all aerological com- 
putations leading to the construction of the upper-level pressure maps. 
For their theoretical interpretation we can therefore safely take our start 
from the hydrostatic equation 
4-16(3) -3/?p5</>. 

The pressure at any level < is then obtained by integration of this equa- 
tion from the level <f> to the upper limit of the atmosphere < , where 
the pressure is zero; thus 

a) 

The pressure is here represented as the weight of the vertical column of 
air of unit cross section extending from the level to the top of the 
atmosphere. 

The local rate of change of the pressure at the level is evidently given 
by the change in weight of the vertical column. It is obtained by partial 
differentiation of (1) with respect to time: 

(2) 

The local rate of decrease of density is, from the equation of continuity 
10*01(5), equivalent to the mass divergence: 

*>P , x , x & / x 

= V*(pV) VH*(PV) + r- (p>V z ). 
Ot OZ 

The last expression on the right gives the mass divergence as the sum of 
horizontal and vertical mass divergence. When this is substituted in 

* J. Bjerknes and J. Holmboe, " On the Theory of Cyclones," Journal of Meteor- 
ology, vol. I, No. land 2, 1944. 



257 



Section 10-07 



equation (2), we have 



The local pressure change at the level < is here represented explicitly as 
the change in the weight of the vertical air column. This change is 
caused in part by horizontal divergence of mass above the level <t> and 
in part by the vertical transport of mass through the base of the column. 

Equation (3), known as the tendency equation, was derived by 
Margules (1904), but most of its practical applications date from 1937. 
The first term on the right will be referred to as the divergence term. In a 
qualitative sense we may visualize positive horizontal mass divergence 
as a horizontal spreading of air, and negative divergence, i.e., conver- 
gence, as a horizontal crowding of air. This makes the physical inter- 
pretation of the divergence integral in the tendency equation quite clear: 
Horizontal convergence increases the mass of air within the vertical 
column and shows up as a pressure rise at the base of the column. 
Horizontal divergence decreases the 
mass of air present within the column 
and makes the pressure fall at the base 
of the column. These effects are shown 
schematically in the left part of fig. 
10-07. 

Equally obvious is the meaning of the 
second term on the right in the tendency 
equation, which will be called the verti- 
cal motion term. An influx of air from 
below into the column increases the 
weight of air inside it and thereby also 
the pressure at its base. Correspond- 
ingly, an outflow of air downwards 
through the base of the column repre- 
sents a loss of weight of the column and 
a decrease of pressure at the base. 
These effects are illustrated by the 
right part of fig. 10-07. 

On a level part of the surface of the earth the vertical motion is zero 
and the tendency equation reduces to 





> * 

> ^ 


%A^ 

^ 


t 


ww 


V 1 ^ 


Pressure rise 


-- 




00 

^ 




00 


^ 

*- 




. 
fr> 
1 


1 




V> { 1 


Pressure fall 


FIG. 10-07. 



(4) 



Section 10-07 258 

All large-scale pressure changes observed on the surface map (except 
special mountain effects) should be explicable in terms of this equation. 
The problem of finding the distribution of horizontal mass divergence 
will be discussed at some length in this chapter. 

A modified form of the tendency equation can be used to compare the 
tendency at the ground (assumed to be horizontal) with the tendency 
at an arbitrary upper level of the same vertical column. When the 
tendency at the level <, equation (3), is subtracted from the tendency at 
the ground, equation (4), we find 



The tendency at the level < may be computed from this equation when 
the tendency at the ground, the vertical motion at the level <, and the 
horizontal mass divergence below the level < are known. The practical 
solution of this problem is discussed further in the next section. 

10-08. The advective pressure tendency. The divergence term in 
the modified tendency equation 10-07(5) may be separated into two 
parts. When the horizontal mass divergence is developed according to 
the formula 10*02(1), we have 



and when the resulting expression on the right is substituted in 10-07(5), 
we find 



(ID " (^ 





It is plausible that horizontal divergence below a fairly low level is 
accompanied by descending motion at that level, and that horizontal 
convergence below the level is accompanied by ascending motion at that 
level. The second and the third term on the right in (1) then have 
opposite signs. The sum of these two terms is, by 10-02(4), 



When the plausible assumption is made that the density of the air parti- 
cles changes only as a result of their vertical displacements, the last inte- 
gral to the right in (2) can be evaluated. It is then found that is 
proportional to the vertical stability of the air, and vanishes for an adia- 



259 Section 10-08 

batic lapse rate. For the stability which normally occurs in the tropo- 
sphere the joint contribution, e, from the second and third terms in (1) to 
the tendency difference can for rough approximations be neglected. 

In the following discussion it will be assumed that the conditions are 
such that the quantity e can be neglected in equation (1). That equa- 
tion then reduces to 



The difference between the pressure tendencies at the ground and at the 
level represents the rate of change in the weight of the intermediate air 
column. According to (3) this change of weight is caused by the hori- 
zontal advection of air from regions of different density in the layers 
below the level <. Thus if the motion has a component from the region 
of denser air, v V//P < and the column becomes heavier. The pressure 
tendency evaluated from (3) will be called the advective pressure 
tendency. 

Rossby has shown that the integral in (3) can be reduced to a simple 
form which can be evaluated from a pilot balloon observation, if it is 
assumed that the wind is geostrophic. The advective pressure change 
which is obtained on the basis of this assumption may be considered as a 
first approximation which is good in a broad and fairly straight current, 
but is less reliable in regions with strongly curved and rapidly changing 
streamline patterns. When the wind is assumed geostrophic, the 
approximate thermal wind equation 8-05(1) is valid; thus 

(4) 20.x--^ 



The isobaric horizontal gradients have here been replaced by the hori- 
zontal gradients, the two being always very nearly equal, due to the small 
inclination of the isobaric surface. When the value of VHP from (4) 
is substituted in the integral of (3), we find 



* 
(pv 

J 



xSv. 



The integral element in the first integral is a scalar triple product (see 
section 614). It changes sign when the cyclic order of the three vectors 
is changed. Since 2Q, Z is a constant vector along the vertical of integra- 
tion, it may be taken outside the integral sign, giving the final expression 
on the right-hand side. 
If p represents the mean density in the layer below the level <, equa- 



Section 10-08 



260 



tion (5) becomes 
(6) 



The integral has a simple geometrical interpretation and can be evalu- 
ated graphically from the hodograph of the wind shear (fig. 8-06a). 
The magnitude of v x v is equal to twice the area 5A swept out by the 
lorizontal wind vector v from the level <j> to the level <t> -f 60, as in 
ig. 10-080. The integral from the ground to the level <t> is thus twice 
the total vector area A swept out by the velocity between the two levels : 
see 11-13. Thus 



(7) 



V 

*/ 



vx6v. 



The vector A is directed along the vertical, upward if the wind turns to 
;he left with increasing height, and downward if the wind turns to the 
ight. Introducing (7) into (6) we find 



(cbi 




FIG. 10-08a. Vector area increment. 




lOmph 



FIG. 10-086. Equivalent sectorial 
area under hodograph. 



n the last expression A has always the same sign as the sense of turning 
if the wind with height. A is an area in the hodograph plane and has 
he dimensions [L 2 T~ 2 ] of velocity square. If mts units are used, the 
quation gives the tendency difference in centibars per second. In 
tactical applications it is more convenient to express the tendency in 
nillibars per three hours, and also to measure the velocity in miles 
>er hour: 1 cb s"" 1 - 1.08 x 10 5 mb (3 hr)- 1 , and 1 (mile hr" 1 ) 2 - 
1.200 m 2 s- 2 . Thus from (8) 



) -[ - 8 - 64 X 



mb (3 hr)- 



261 . Section 10-08 

where A is expressed in (miles hr"" 1 ) 2 . The formula is used primarily to 
compute the advective pressure tendency on an upper-level map, for 
instance the 10,000-ft map, from a pilot balloon observation and the 
known tendency at the ground. If we assume the value 922 m 3 tT 1 for 
the mean specific volume of the column from the ground to 10,000 ft, 
i.e., the value of at p = 850 mb, T = 273K, the formula takes the form 



*)-(*) - 6 ' 83 x 10- 3 4 sin * mb (3 hr)- 1 



The practical procedure in computing the 10,000-ft advective pressure 
tendency from this formula is shown schematically in fig. 10-086. The 
hodograph of the wind distribution along the vertical from the ground to 
the 10,000-ft level as obtained from a pilot balloon observation is plotted 
on a polar diagram. The points for the surface and each 1000 ft of ele- 
vation are marked 5, 1, 2, , 10. The variation of the wind in the sur- 
face layer (schematically shown as an Ekman spiral) is primarily caused 
by friction. The density advection in this layer is not in accord with 
the simple formula (10), which was derived on the assumption that the 
wind is geostrophic. Although the advection in the frictional layer may 
be of some consequence, its contribution is omitted from the computa- 
tion, since no simple technique is at hand for its evaluation. A rough 
average value for the geostrophic wind level over land is 3000 ft. The 
contribution to the advective pressure change between the geostrophic 
level and the 10,000-ft level is measured by the area under the hodograph 
between these levels. Instead of evaluating this area directly, we may 
evaluate the area of an equivalent circular sector. The circular arc has 
been drawn such that the two triangular areas are equal. The area of 
the circular sector is -%6v 2 , where is the sectorial angle, measured in 
radians, and v is the mean speed of the layer measured in miles per hour. 
Introducing this value in the formula (10), we find 



(11) ( I -(T-) - 3.42 x l<T 3 Ov 2 sin ^ mb (3 hr)- 1 . 
\&/o \&/io 

The area %0v 2 can be read off the diagram directly if lines of constant 
value of 6v 2 are drawn in the polar diagram. One such line is shown in 
fig. 10-08c. For a given latitude the tendency difference is directly pro- 
portional to the area. The lines of constant Ov 2 may therefore be drawn 
and labeled directly for unit values of the tendency difference, as shown 
in fig. 10-08d. The diagram has been computed for the latitude 40. 
In practical use the diagram is drawn on transparent paper, or prefer- 
ably a thin sheet of celluloid. When the pilot balloon hodograph has 
been plotted on a regular polar diagram the transparent area computer, 



Section 10-08 



262 



fig. 10-08d, is placed on top of it. If, in the northern hemisphere, the 
wind turns to the right with increasing height (warm air advection), the 
base line of the computer is placed along the 10,000-ft velocity vector. 



= const 






FIG. 10-08c. Line of constant sectorial area. 

The area and hence the tendency difference is negative in this case. If 
the wind turns to the left (cold air advection), the base line of the area 
computer is placed along the 3000-ft velocity vector. The area and 
hence the tendency difference is positive in this case. 

6 mb 




70 60 50 40 30 25 
ft, = 1.5 1.3 1.2 1.0 0.8 0.7 

FIG. 10-08d. Area computer for advective pressure change. 

If the computer is used for a station in another latitude <p, the values 
indicated by the computer must be multiplied by the factor 
k v = sin ^>/sin 40. The values of this correction factor for a number of 
latitudes are listed below the base line of the computer. 

It should be noted that the 10,000-ft tendency as obtained by this 
method is only a first crude approximation. The method is based on the 
assumptions that the quantity e in (2) is zero; that the wind is geo- 
strophic; that advection in the frictional layer can be neglected ; and 
that 1/p is near 922m 3 1" 1 . 



263 Section 10-09 

10-09. Relation between the horizontal divergence and the field of 
pressure. The equation of motion implicitly contains the relation 
between the field of motion and the pressure field. By making suitable 
assumptions it is possible to estimate the distribution of the horizontal 
divergence directly from the horizontal pressure field. 

In the layer of frictional influence near the surface of the earth the 
winds have a systematic component across the isobars from high to low 
pressure (chapter 9). Every " low " is then a region of horizontal mass 
convergence and every " high " a region of horizontal mass divergence, 
within the layer of friction. However, the layer of friction represents 
only about one-twentieth of the weight of the atmosphere, and any hori- 
zontal divergence in this layer may easily be overcompensated by hori- 
zontal divergence in the remaining nineteen-twentieths of the atmos- 
phere. 

In the free atmosphere, above the layer of frictional influence, the 
wind is in the first approximation geostrophic, blowing parallel to the 
isobars of the pressure map. Furthermore, at any given latitude 
the geostrophic wind is inversely proportional to the distance between 
the isobars. We can therefore think of the isobars of a constant-level 
pressure map as representing the horizontal motion of the air in the 
same level. The air flows along " isobaric channels " covering the strips 
between successive isobars. Where the channel is wide, the air flows 
slowly, whereas in the narrow parts of the channel the air flows rapidly. 
The product of the density p, the wind speed v, the channel width 8n, 
and the constant channel depth 8z defines what may be called the trans- 
port capacity dF of the isobaric channel : 

(1) 8F=pvdndz. 

The transport capacity is practically constant for a reasonably straight 
channel which runs about west-east along a circle of latitude. 

If the transport capacity of isobaric channels were constant all over 
the map there would be no convergence or divergence to produce pres- 
sure changes. Every isobaric channel would be like a well-regulated 
river with constant transport all along, so that no local accumulations or 
depletions would occur. However, this is fulfilled only approximately 
in the atmosphere, because the geostrophic wind is only a first approxi- 
mation to reality. Actually the isobaric air channels change their trans- 
port capacity somewhat from point to point. The places of minimum 
transport capacity will then be " bottlenecks " in the flow of air. The 
air approaching a bottleneck will crowd horizontally and cause a longi- 
tudinal mass convergence in the current, whereas beyond the bottleneck 
there will be longitudinal mass divergence. To complete the list of 



Section 10-09 



264 



shortcomings of the geostrophic wind, the air does not follow strictly 
along the isobaric channels but overflows slightly from the one to the 
other. It overflows toward lower pressure when the air speeds up, and 
toward higher pressure when the air slows down. This also may cause 
horizontal mass divergence which will be referred to as transversal mass 
divergence. For brevity, we shall hereafter in chapter 10 write " diver- 
gence " instead of " mass divergence." 

In the following, the longitudinal and transversal divergence will be 
evaluated quantitatively, or estimated qualitatively, for selected simple 
patterns of flow under the two general headings: (i) wave-shaped flow 
patterns, and (ii) closed flow patterns. These flow patterns, of course, 
correspond to wave-shaped isobar patterns and closed isobar patterns 
respectively. The former are mainly observed in the upper levels; the 
latter, in the lower levels of the traveling cyclones. The results from the 
study of these fundamental patterns will thus ultimately throw some 
light on the mechanism of the composite cyclonic disturbances. 

10- 10. Longitudinal divergence in wave-shaped isobar patterns. 

The type of wave-shaped isobar pattern to be studied is shown sche- 
matically in fig. 10'lOa. The amplitude of the wave disturbance has a 
maximum in the middle of the pattern and tapers off to the north and to 
the south of this latitude. This resembles the usual pressure distribu- 

N 



W 




E 



FIG. 10-lOa. Wave-shaped isobar pattern. 

tion on an upper-level map above a moving cyclone in the latitudes of 
the westerlies. In order to facilitate the discussion it has been assumed 
that the isobars are symmetric with respect to the trough lines and the 
wedge lines, and that these lines are straight lines with north-south orienta- 
tion. 

It follows from the symmetry of the pattern with respect to the trough 
lines and the wedge lines that these lines are the places of maximum or 



265 Section 10-10 

minimum transport capacity of each isobaric channel. To estimate the 
distribution of longitudinal divergence, it is thus sufficient to evaluate the 
transport capacity of the isobaric channels at their southern bends 
(where they intersect the trough lines) and at their northern bends 
(where they intersect the wedge lines). From the place of minimum to 
the place of maximum transport capacity (along the direction of the 
current) there will be longitudinal divergence, and from the place of 
maximum to the place of minimum transport capacity there will be 
longitudinal convergence. 

The trough lines and the wedge lines are not only the places of maxi- 
mum or minimum transport capacity of the isobaric channels but are also 
the places of maximum or minimum speed. At such points the flow is 
gradient flow along the isobars, so that in the normal component equa- 
tion of motion, 

7-13(5) K H v 2 + 2to> sin p = - - ~ > 

p on 

8n is measured normal to the isobar. Therefore 8n is the width of the 
isobaric channel bordered by the isobars p and p -f 8p. In the following 
the horizontal curvature K H will be denoted by K. No ambiguity will 
arise, for only horizontal curvatures will be used. 

Multiplying equation 7-13(5) by pdndz and introducing the notation 
67? for the transport capacity from 10-09(1), we have 

(1) 8F(Kv + 2ft sin <p) = -dpdz = const. 

The channel depth 62 and the pressure difference across the channel 
dp are both constant all along the isobaric channel. The transport 
capacity at the places with gradient flow is accordingly proportional 
to the reciprocal of the quantity 
(Kv+2Q sin tp). The distribution 
of longitudinal divergence is there- 
fore obtained by comparison of the 
values of this quantity at the places 

where the isobaric channel inter- ^ in < n , T , . , f r 

, f . f f , FIG. 10-106. Isobanc channel from 

sects the trough line and the wedge thc pattern in fig 1(MOa- 

line. 

Fig. 10'lOft shows an arbitrary isobaric channel in the pattern of 
fig. 10-10a, bordered by the isobars with the pressures p and p + bp. 
The channel has its cyclonic bend at the latitude <p where it intersects 
the trough line, and its anticyclonic bend at the latitude <p' where it 
intersects the wedge line. Obviously the relation <p f > v holds in all 
cases. In the following all the quantities at the cyclonic bend will be 




Section 10-10 266 

denoted by unprimed symbols, and the corresponding quantities at the 
anticyclonic bend by primed symbols. Note that 8p < 0. 

The transport capacities dF' and dF at the two bends must satisfy one 
of the following three conditions 

(2) IF* | dF. 

The upper condition states that the bottleneck in the current is at the 
cyclonic bend of the isobaric channel. West of this bend air accumu- 
lates and causes longitudinal convergence; east of the bend there will be 
longitudinal divergence. The middle condition states that the transport 
capacities are the same at the two bends, so the longitudinal divergence 
is zero. The lower condition states that the bottleneck in the current 
is at the anticyclonic bend of the isobaric channel, so the flow has longi- 
tudinal convergence to the west of this bend and longitudinal divergence 
to the east of it. 
According to equation (1) the conditions (2) can be written 

(2') Kv + 20 sin <p | K'V' + 20 sin <p 

or, after rearrangement of the terms, 



(3) Kv K'V' ^20 (sin <p' sin <p) 40 cos - sin - 

JL t 

The angle (<p 7 -f <p)/2 is the central latitude <f>, around which the isobaric 
channel winds, and trie angle (<?' <p)/2 is half of the difference in lati- 
tude between the northern and the southern bends of the channel. The 
latter may be called the angular amplitude of the isobar and is denoted by 
a p ; thus 



fA\ - 

(4) <f> = i a p = 



With these notations introduced, (3) takes the form 
(5) Kv - K'v* | 40 cos sin <r p . 

This formula contains the horizontal curvatures of the air trajectories at 
the southern and the northern bends of the isobaric channel. These 
curvatures depend upon the shape of the streamlines and the speed of 
propagation of the wave disturbance. 

The horizontal curvature K of the path is related to the horizontal 
curvature K s of the streamline by the equation 

7-23(3) j& 



267 Section 10-10 

where cty/dtf is the local rate of turning of the wind. At any identifiable 
point in the moving pressure pattern the relative rate of turning of the 
wind is given by 

10 ' 05(2) f=! +c ' v *- 

Let the identifiable point be the intersection between the trough line (or 
the wedge line) and one of the circles of latitude. The velocity c is then 
a zonal vector, tangential to the path at the troughline, and so 
cV^ = c(<ty/ds). Furthermore the wind is always zonal at the trough 
line, so the relative turning of the wind, d#l//dt, is zero. Accordingly, the 
local rate of turning of the wind at any point on the trough line or the 
wedge line is 

ty W 

= - c-~ = -cK S - 
dt ds 

d^/ds is by definition (see section 7-23) the horizontal curvature of the 
streamline. Substituting this expression for the local turning into the 
general formula 7-23(3), we have at the two bends 



Let v be the mean zonal wind in the isobaric channel, and let At; be 
the half difference of the speeds at the northern and southern bends. 
Thus: 



- ~ 

(7) v = j- > A0 -- 

Introducing these in (6), we find 

Kv= K s (v-c)- 
K'v' - KS(V - c) 
The difference between these expressions is 

(8) Kv - K'v' - (K a - K' a ) (v - c) - (K s + K' s ) Av. 
Substituting this in (5) and solving for v - c, we have finally 

f*\ A > 40 cos y sin (T P Jgfl + JC& 

(9) '~ e * K S -K' S +1^W S ^ 

The expression on the right has the dimensions of a velocity. It is 
completely determined by the geographical location and geometry of the 



Section 10-10 268 

pattern, and by the difference of wind speeds at the northern and south- 
ern bends of the isobar. We shall in the following refer to this quantity 
as the critical speed and denote it by v c \ thus 

MAN 4Q cos ? sin <r p K s + K' s 

(10) v c f + - =7 Ai>. 

AS AS AS AS 

With this notation introduced, equation (9) takes the form 

(11) v-c\v c . ' 

i) c is the mean value of the zonal speed relative to the moving wave 
pattern, and will be called more briefly the relative zonal wind. 

The three conditions (11) are equivalent to the three conditions (2) 
taken in the same order : Thus, if the relative zonal wind is supercritical, 
that is, greater than the critical speed, the bottleneck in the current is at 
the cyclonic bend of the isobaric channel, so the flow has longitudinal 
convergence to the west of the trough and longitudinal divergence to the 
east of the trough. If the relative zonal wind is critical, that is, equal to 
the critical speed, the transport capacities are the same at the two bends, 
so the longitudinal divergence is zero. If the relative zonal wind is sub- 
critical, that is, less than the critical speed, the bottleneck in the current 
is at the anticyclonic bend of the isobaric channel, so the flow has longi- 
tudinal convergence to the east of the trough and longitudinal divergence 
to the west of the trough. 

The expression (10) for the critical speed becomes much simpler when 
the curvature is numerically the same at the cyclonic and the anti- 
cyclonic bends, that is, when KS = -K' s . The second term on the 
right side of equation (10) is then equal to zero, and the critical speed is 

, N 212 cos sin or p 

(12) v c = 

&s 

So far no analytical expression has been specified for the streamlines or 
the isobars. The only restriction in the choice of isobaric pattern pre- 
scribes that the isobars should be symmetrical with respect to the north- 
south trough lines and wedge lines. The critical speed may be deter- 
mined directly on the weather map from (10). If the curvatures at the 
two bends are numerically equal, we may use formula (12). This for- 
mula may be simplified still further when the streamlines are assumed to 
be sine curves. 

10-11. Critical speed in sinusoidal waves. In order to treat the 
streamlines as simple sine curves, it is necessary to consider the surface 
of the earth as flat and the latitude circles as straight parallel lines in the 



269 



Section 1(M1 



region between the two bends. A sinusoidal streamline can then be 
drawn winding between the latitudes <p and ^', as shown in fig. 10*11. 
In a standard system of coordinates with the origin at the point where 
the streamline intersects the central latitude at the time t = 0, this 
streamline has the equation 



(1) 



y = AS sin k(x- ct). 




FIG. 10-11. Sinusoidal streamline. 

Here A $ is the amplitude and c is the speed of the wave. If L$ is the 
wave length, k = 2ir/Ls is called the wave number and represents the 
number of waves in the linear interval of 2ir length units. 

If the latitudinal amplitude of the streamline is denoted by 0-5, the 
linear amplitude A s on the horizontal plane is defined by projection as: 



(2) 



AS = a tan 



The linear wave length L$ can be expressed by the angular wave length 
of longitude X# as follows: 



cos 



The angular wave length defines an angular wave number n 
which gives the number of waves along the total circumference of the 
latitude circle. The relation between the linear wave number k and the 
angular wave number n is 



(3) 



2* 
L s 



2-ir 



n 



\sa cos q> a cos 



The curvature of the streamline is by definition the angular turn of its 
tangent per unit arc length along the streamline; thus 

(4) K s - 



Section 10-11 270 



If \f/ is the angle between the tangent and the x axis, then 

by 

tan v = 
dx 

Differentiating this formula with respect to the arc length 5, we have 



, 

cos 2 ^ 65 

At the southern and northern bends, where the streamline is parallel to 
the x axis, cos 2 ^ = 1 and 65 = dx. Therefore from (4) and (5) we have 



When (1) is differentiated twice with respect to #, we find for the curva- 

ture of a sinusoidal streamline at the two bends 

(6) K s = -k 2 A s sin k(x - a) - k 2 A 8 . 

In the last expression on the right the positive sign should be used at the 
southern bend and the negative sign at the northern bend. 

By substituting in (6) the values for ^4^ from (2) and for k from (3), 
the curvature at the southern bend becomes 

n 2 tan <TS 

Ks - - g -- 

a cos 

When this expression for Kg is introduced in 1(MO(12), we find for the 
critical speed in a sinusoidal flow pattern the expression 

212 a cos 3 <p sin <r p 



(7) v c 



o 
n 2 tan 



This formula was derived by Rossby (1939) from the principle of con- 
servation of vorticity (see 12-05). It will follow from (7) and 12-05(12) 
that the amplitude factor sin o-^/tan a p is equal to one. This would 
indicate that the streamline amplitude is slightly larger than the isobar 
amplitude. However, in section 12-06 it will be proved that the two 
amplitudes are equal at the level of zero longitudinal divergence. This 
apparent error in the formula (7) comes from the fact that it has been 
derived by combining spherical and plane methods. The basic formula 
10-10(12) is exact on a spherical level for a wave-shaped isobar with the 
same streamline curvature at the northern and the southern bends. 
This formula cannot be specialized to a curve in a " plane level " with- 
out causing a slight degree of inconsistency. 

The critical speed in a sinusoidal flow pattern is thus a function of the 
latitude and the angular wave number. The critical speed is large for 



271 Section 10-12 

long waves in low latitude, and small for short waves in high latitude. 
Some values of the critical speed, as computed from (7), are given in 
table 10-11. 

TABLE 10-11 
v e = (2flacos 3 ^>)/w 2 (M s"" 1 ) TABULATED 





n 


9 


2 


3 


6 


10 


20 


70 


9.3 


4.1 


1.0 


0.4 


0.1 


60 


29.0 


12.9 


3.2 


1.2 


0.3 


50 


61.6 


27.4 


6.8 


2.5 


0.6 


40 


104 


46.3 


11.6 


4.2 


1.0 


30 


151 


67.0 


16.8 


6.0 


1.5 





232 


103 


25.8 


9.3 


2.3 



For long waves it seems unwarranted to consider the earth as flat and 
to neglect the curvature of the latitude circles. No detailed discussion 
of this question will be given here. It may suffice to mention that 
waves spanning over one-half to one-third of the circumference of the 
earth must be treated by spherical methods. Practical synoptic cases 
are more likely to deal with wave numbers around n = 6 or more (wave 
lengths 60 of longitude or less), and in such cases the difference between 
the flat and the spherical treatment seems to be insignificant. 

10*12. Transversal divergence in wave-shaped isobar patterns. We 
now turn our attention to that part of the horizontal divergence which 
is caused by the overflow of mass from the one isobaric channel to the 
other, i.e., the transversal divergence. Fig. 10-12 shows the same ideal- 
ized pressure pattern which was discussed in section 10-10. Aqualitative 
estimate of the transversal divergence in this pressure pattern is obtained 
by considering the inflow and outflow of mass across the northern straight 
isobar p = po 4 (cb), and across the southern straight isobar p p Q . 

Suppose for a moment that the isobar pattern in fig. 10-12 is station- 
ary. Particles at the northern edge of the pattern will then have their 
maximum speed while passing the longitudes of the wedges and minimum 
speed while passing the longitudes of the troughs. In order to change 
speed in that rhythm the particle must have a component toward high 
pressure while it slows down, i.e., from the wedge to the trough (see 
section 7-15). And it must have a component toward low pressure 
while it speeds up, i.e., from the trough to the wedge. The flow at the 
northern edge must therefore be as qualitatively indicated by the stream- 



Section 10-12 



272 



line (dotted line) in the diagram. The analogous analysis along the 
southern straight isobar leads to a streamline with opposite phase. 
There is thus outflow from the region east of the trough, both across the 
northern and the southern edge of the isobar pattern, and therefore 
transversal divergence from this region. And there is transversal con- 
vergence into the region west of the trough. 



N 



P.- 3 

Po-2 
W 

Po-1 




Po-4 



Po-2 
E 

Po-1 



FIG. 10-12. Flow pattern for the isobar pattern in fig. 10-lOa. 

If the pressure system is moving, the same rule holds, provided that 
the system does not move eastward faster than the mean speed of the air, 
i.e., provided that the relative zonal wind is from the west. If the pres- 
sure system moves eastward with the mean zonal speed, the relative 
zonal wind is zero, arid the transversal divergence of the flow pattern is 
zero. If the pressure system moves eastward faster than the air, the 
relative zonal wind is from the east; the particles at the northern edge 
of the frame slow down east of the trough (while being overtaken by the 
trough), and they speed up to the west of the trough (while being over- 
taken by the wedge). The streamlines at the northern and southern 
edges of the frame will then have the opposite phase from those shown in 
fig. 1042 and consequently give transversal convergence into the region 
east of the trough and transversal divergence from the region west of it. 
These rules may be summarized as follows: The transversal divergence 
and the relative zonal wind have the same sign in the region to the east of the 
trough, and have opposite signs in the region to the west of the trough. 

The above rule only gives the sign of the transversal divergence. The 
mean divergence over a given area is obtained by dividing the total out- 
flow from that area by the area. The areas of outflow and inflow in the 
wave pattern, fig. 10-12, are proportional to the meridional distance 
between the northern and southern straight isobars. The transversal 



273 Section 10-13 

divergence is therefore proportional to the reciprocal of the width of the 
pattern and is zero for a pattern of infinite lateral extent. All the isobars 
then have the same shape, and the longitudinal divergence is the total 
horizontal divergence. 

10-13. Total horizontal divergence associated with wave-shaped 
isobar patterns. The total horizontal divergence is the sum of the 
longitudinal and the transversal divergence. The following three alter- 
natives may occur : 

1. v c > v c : The relative zonal wind is from the west and is greater 
than the critical speed ; the longitudinal and the transversal divergence 
are both positive to the east of the trough and are both negative to the 
west of the trough. 

2. v c < 0: The relative zonal wind is from the east; the longi- 
tudinal and the transversal divergence are both negative to the east of 
the trough and are both positive to the west of it. In both cases 1 and 2 
the two parts of the horizontal divergence have the same sign at any point 
of the flow pattern. 

3. v c > v c > 0: The relative zonal wind is from the west and is 
smaller than the critical speed; the longitudinal divergence is negative to 
the east of the trough and positive to the west of the trough ; the trans- 
versal divergence is positive to the east of the trough and negative to the 
west of it. The two parts of the horizontal divergence then counteract 
each other throughout the field. For a certain value of the relative 
zonal wind less than the critical speed but larger than zero the longi- 
tudinal divergence is balanced by transversal convergence and the total 
horizontal divergence is zero. 

The distribution with height of the horizontal divergence in an actual 
synoptic situation can be estimated with the aid of the above rules for 
longitudinal and transversal divergence. In general, the strength of the 
zonal circulation and hence also of the relative zonal circulation increases 
with height. In the region to the east of every trough and to the west of 
every wedge there is longitudinal divergence above the level where the 
relative zonal wind has the critical speed, and there is longitudinal con- 
vergence below that level. In the same region there is transversal 
divergence above the level where the relative zonal wind is zero, and 
transversal convergence below that level. Between the level of zero 
longitudinal divergence and the level of zero transversal divergence the 
two effects will have opposite signs. At some intermediate level the 
longitudinal and the transversal divergence are equal and opposite, and 
the total horizontal divergence is zero. 

The distribution both of the longitudinal and the transversal diver- 



Section 10-13 274 

gence depends upon the relative zonal wind and hence upon the speed 
of the wave. The speed of the wave prescribes the pressure tendency. 
The pressure tendency is, from the tendency equation 10-07(3), a direct 
consequence of the distribution of horizontal divergence. Thus the 
horizontal divergence, the speed of the wave, and the pressure tendency 
are interdependent : The wave will travel with such a speed that the pressure 
tendencies arising from the displacement of the pressure pattern are in 
accordance with the field of horizontal divergence. 

The three-dimensional structure of the wave disturbance may be 
analyzed from this point of view. The conditions are fundamentally 
different in a barotropic and in a baroclinic current. Only the rather 
unreal barotropic case is as yet accessible to investigation by rigorous 
dynamical analysis. In order to gain the greatest possible experience 
we shall first examine the barotropic wave. Later we shall proceed to 
the qualitative analysis of the much more complicated conditions in the 
real atmospheric waves, which are always baroclinic. 

10-14. Barotropic waves in a westerly current. To simplify matters 
we shall assume that the wave pattern has infinite lateral extent, so that 
all isobars have the same shape. The transversal divergence is then 
always zero, and the total horizontal divergence is equal to the longi- 
tudinal divergence. 

In a barotropic current the isothermal surfaces coincide with the iso- 
baric surfaces. The dynamic thickness of each isobaric layer is therefore 
constant throughout. The slope of all isobaric surfaces is then the same 
along any given vertical, and the geostrophic wind has no vertical shear 
(see section 8-03). The strength of the zonal circulation and the shape 
of the streamline pattern of the wave are therefore the same at every level. 

The pressure tendency at the ground indicates whether the wave 
moves or not. If we assume that the surface of the earth is flat this 
tendency is 



uu 

w\ _ f 

,5//o J 



10-07(4) (^j -- / V/f(pv)5. 

\O//o J 



The sign of the mass divergence is determined from the three conditions 
10-10(11) v-c^v c . 

The mean zonal wind v has the same value at all levels in a barotropic 
current; the speed c of the wave is characteristic of the entire wave and 
therefore independent of height; the critical speed v c is determined by the 
wave length and so is also independent of height. Thus the mass diver- 



275 



Section 10-14 



gence has the same sign throughout any vertical column. Fig. 10-14 
illustrates the barotropic wave for each of the three conditions 10-10(11). 
The diagram to the left represents a wave for which the relative zonal 
wind is supercritical, 

(1) v-c>v c . 

The bottleneck in the current will then be at the cyclonic bend of every 
isobaric channel. Air is accumulated at all levels to the west of the 
trough, and is depleted at all levels to the east of the trough. At the 



v-c > v 




+ 00 + 

FIG. 10-14. Propagation of barotropic wave. 

ground the pressure is therefore rising to the west of the trough and fall- 
ing to the east of it. So the wave moves to the east (c> 0), and hence 
from (1) v > v c . 

The center diagram represents a wave for which the relative zonal 
wind is critical, 

(2) v - c = v c . 

The transport capacities are the same at the two bends of each isobaric 
channel. The entire flow is non-diverging and the pressure tendency 
at the ground is zero. So the wave is stationary (c = 0), and hence 
from (2) v = v c . 

The diagram to the right represents a wave for which the relative 
zonal wind is subcritical, 

(3) v-c< v c . 



Section 10-14 276 

The bottleneck in the current will then be at the anticyclonic bend of 
every isobaric channel. Air is accumulated at all levels to the east of 
the trough and is depleted at all levels to the west of the trough. At the 
ground the pressure is therefore rising to the east of the trough and fall- 
ing to the west of it. So the wave moves to the west (c < 0), and hence 
from (3) v < v c . 

The above results may be summarized as follows: 

(4) v - c | v c , 

(5) c 1 0, 

(6) o | v c . 

We have shown that the three conditions (4) read from top to bottom 
imply the conditions (5) taken in the same order. The conditions (4) 
and (5) combined give the conditions (6) . It is readily seen that any one 
of (4), (5), (6) implies the other two. Thus from (5, 6) the barotropic 
wave moves eastward when the mean zonal wind is supercritical; it is 
stationary when the mean zonal wind is critical; it moves westward when the 
mean zonal wind is subcrilical. 

For waves of finite width the transversal divergence is different from 
zero. The only reformulation of the above rules is, then, that the 
stationary, non-diverging wave occurs for a smaller value of the zonal 
wind than the critical speed. 

The tendency at upper levels is given by the complete tendency 
equation 



10-07(3) ^ - - JV/,-Gv)8* +(.)* 



We have shown that the divergence has the same sign throughout any 
given atmospheric column, so the divergence term has the greatest magni- 
tude at the ground and decreases with height. The vertical motion 
term is zero at the ground and increases in magnitude with height. In 
the special case of a homogeneous current (p independent of p) the pres- 
sure tendency would be the same at all levels. Here the decreasing con- 
tribution from the divergence term with height is exactly balanced by 
the increasing contribution from the vertical motion term. In the more 
general barotropic case where the density decreases with height, both 
the pressure amplitude and the total tendency decrease upward. In this 
case the contribution from the divergence term must decrease more 
rapidly with height than the contribution from the vertical motion term 
increases. 

The zero isallobars at all levels coincide with the trough lines and the 




277 Section 10-15 

wedge lines. This means that the amplitude of the pressure wave does 
not change with time; in other words, the barotropic wave is a stable 
wave, which moves without any changes in its internal structure. This 
result is in complete accord with classical wave theory. The only 
destabilizing factor, shear, is absent in the barotropic wave, so the wave 
must be stable. 

10*15. The relative streamlines. We shall now examine waves in a 
baroclinic westerly current. In a barotropic current the isotherms coin- 
cide with the isobars at all levels, but in a baroclinic current they will 
generally intersect the isobars. Therefore the structure of the wave will 
be different from level to level, and will also change with time. 

The internal structure of a ^^ 

moving wave is deformed by 
the relative wind, that is, the 

wind as seen by an observer who v c> 

moves along with the wave. 
The streamlines of this relative 
wind are the relative stream- 
lines. Let v be the wind and c 

the velocity of propagation of ^ ^^ 

Al _ , * i ,- FIG. 10-15a. Streamline and relative 

the wave. Ihe wind relative streamline. 

to the wave is then v - c. The 

relative streamlines are the streamlines of the velocity field V c. See 

fig. 10-15a. 

At the trough lines and the wedge lines of the streamline pattern both 
the real wind and the relative wind are zonal. These meridians are 
therefore also the trough or wedge lines of the relative streamline pattern. 
So both patterns have the same wave length, but their amplitudes are in 
general different. The streamlines are by definition everywhere tan- 
gential to the velocity. Let dy$ and 8yR be the respective meridional 
increments on the streamline and on the relative streamline, correspond- 
ing to the same zonal increment 8x. The differential equations for the 
streamlines and the relative streamlines are then respectively 



dx 




8x v x - c 
Taking the ratio of these equations, we have 



Section 10-15 



278 



The ratio on the right side in (1) is constant in the special case where, 
throughout the level, the zonal wind component v x has the constant 
value v of the actual wind at the trough line and at the wedge line. 
Equation (1) may then be integrated. If the integration is taken from 
the inflection point (the central latitude) to the wedge line, we find 

(2) 4*.Llf. 

AR V 

When the wind speed v at the trough line is different from the wind 
speed v' at the wedge line, the ratio between the two amplitudes is not so 
simple as indicated by (2). However, if the difference v f - v is small 
compared to the mean speed v, it can be shown that the amplitude ratio 
has the approximate value 



(3) 



As 

AR 



v-c 



Thus, in a wave where the variation in the zonal wind is not too large, the 
ratio of the streamline amplitude to the relative streamline amplitude 
is the same as the ratio of the mean relative zonal wind to the mean zonal 
wind. 

Fig. 10-156 shows the streamline (full line) and the relative streamline 
(broken line) for three different values of the speed. In all three cases 
the streamline is the same. The relative streamline is obtained from 

the formula (3), or directly by inspec- 
tion of the velocity vector diagrams in 
the figure. In the upper diagram the 
wave is stationary (c = 0), and the 
relative streamline coincides with the 
streamline, AS = AR. In the middle 
diagram the wave moves toward the 
right with a slower speed than the 
air. The relative streamline is here 
in phase with the streamline, but 
the amplitudes differ in the sense 
AS < AR. In the lower diagram the 
wave moves to the right with a greater 
speed than the air. In this case the 

relative zonal wind is negative, so the amplitudes A s and AR have 
opposite signs. The relative streamline is 180 out of phase with the 
streamline. 

A wave moving without change of shape appears stationary to an ob- 
server moving along with the wave. For such a wave the relative 





FIG. 10-156. 



279 Section 10-16 

streamlines are also the relative paths. The relative streamlines thus 
indicate the true meridional displacements of the air particles. 

10*16. Stable baroclinic waves. A wave whose shape and internal 
structure do not change during its propagation will be called stable. 
We shall investigate whether stable waves can exist in a baroclinic 
current, and study the three-dimensional structure of such waves. 

We shall assume that the motion is nearly horizontal and that the 
temperature of individual particles is conserved. If the relative stream- 
lines were to intersect the isotherms, the temperature field relative to 
the moving wave would be deformed and the wave would not be stable. 
Therefore in a stable baroclinic wave the relative streamlines must coincide 
with the isotherms. 

This rule implies that the isotherms have the same shape at all levels 
if the wind is assumed to be geostrophic. By the thermal wind equation, 
8*05(1), the shear of the geostrophic wind is directed along the horizontal 
isotherms. Since, now, a stable wave moves with the same speed at all 
levels, the shear of the wind is also the shear of the relative wind. So 
the shear of the relative wind at any level is directed along the isotherms 
and, hence, along the relative streamlines at that level. Thus the rela- 
tive wind does not turn with height, so the relative streamlines have the 
same shape at all levels. 

This property of the relative streamlines, combined with the relation 
between the relative and the real streamlines, makes it possible to pre- 
dict the whole three-dimensional structure of a stable baroclinic wave. 
The main features of such a wave are shown schematically in fig. 10-16. 
The central part of the diagram represents horizontal maps at four 
selected levels, with one streamline and one relative streamline drawn in 
each map. The relative streamlines (broken lines) have been given the 
same shape at all levels. The wave is assumed to move toward the east. 
The speed of the wave is the same at all levels. But on account of the 
baroclinic temperature field (warm to the south and cold to the north) 
the zonal wind increases with increasing height, as indicated in the left 
part of the diagram. At some level the speed of the zonal current must 
be equal to the speed of the wave. At that level the relative zonal wind 
is zero; from 10-15(3), AS = 0, so the streamlines are straight. Below 
that level the air is being overtaken by the wave and the streamlines are 
180 out of phase with the relative streamlines. Their amplitude 
increases with increasing depth below the level of straight streamlines. 
Above the level of straight streamlines the air overtakes the wave; the 
streamlines are in phase with the relative streamlines; and AS < AR. 

When the streamlines are identified as isobars, it is readily seen that 



Section 10-16 



280 



the change of the pressure distribution from level to level is qualitatively 
in accord with the hydrostatic equation. The warm trough at low levels 
vanishes with height and is surmounted by a warm pressure wedge, whose 
isobar amplitude increases with height. Correspondingly, the cold 
pressure wedge at low levels vanishes with height, and is surmounted by 
a cold trough whose isobar amplitude increases with height. 

We can now estimate the transversal, longitudinal, and total mass 
divergence in the various parts of the wave. This estimate, as applied to 
the column A A , is shown qualitatively in the three graphs to the right in 
fig. 10-16. 

The transversal divergence is zero at the level where the relative zonal 
wind is zero (v = c). Above that level the relative zonal wind is posi- 
tive, so there is transversal divergence to the east of the trough and trans- 



Trans Long Total 




\ X 





FIG. 10-16. Distribution of divergence in a stable baroclinic wave. 

versal convergence to the west of it. Below the level where v=* c, the 
relative zonal wind is negative, so there is transversal convergence to the 
east of the trough and transversal divergence to the west of it. There- 
fore the column A A has transversal divergence both above and below 
the level where v = c. 

The longitudinal divergence is zero at the level where the relative 
zonal wind is critical (v - c = v c ). Above that level the relative zonal 
wind is supercritical, and the bottleneck is at the cyclonic bend. So 
there is longitudinal divergence to the east of the trough and longi- 
tudinal convergence to the west of it. Below the level where v c - v c 
the relative zonal wind is subcritical, and the bottleneck is at the anti- 
cyclonic bend. So there is longitudinal convergence to the east of the 



281 Section 10-17 

trough and longitudinal divergence to the west of it. Therefore the 
column AA has longitudinal divergence above the level where v - c = v c 
and also below the level where v = c. Between these levels the column 
A A has longitudinal convergence, for the relative wind is subcritical, 
and the column is to the east of the trough. . 

The total horizontal divergence is obtained by addition of the trans- 
versal and longitudinal parts. For the column AA it is positive at all 
levels except in a layer whose base is the level where v = c, and whose 
top is below the level where v - c = v e . 

If the level is interpreted as a flat part of the earth's surface, it seems 
rather likely that divergence may dominate at A A , and convergence at 
BB. This would mean pressure fall at the base of column AA and 
pressure rise at the base of BB. This again would make the pressure 
wave at the ground move westward, contrary to our initial assumption. 
The only place and time that a trough like that shown in fig. 1046 
could move eastward would then be while the trough is over a mountain 
range. The downward velocity on the lee side of the mountain could 
then be strong enough to overcompensate the effect of convergence in the 
column BB above and produce pressure fall ahead of the trough. Like- 
wise the upslope wind on the west side of the mountain could overcom- 
pensate the effect of the divergence in column AA and produce pressure 
rise behind the trough. 

If now the wave pattern moves eastward, columns like AA with a pre- 
ponderance of divergence will arrive over the lee side of the mountain. 
The influence of downward motion and divergence will add up in the 
tendency equation, and a strong pressure fall at the ground will result, 
which will continue on the lee side till columns with a preponderance of 
convergence arrive after the passage of the upper-level trough. This is 
equivalent to a deepening of the low-level trough over the lee slope of the 
mountain and a belated arrival of the following high-pressure wedge to 
the same region. 

10*17. The first formation of the baroclinic wave. We are now 
ready to take up the fundamental problem in J. Bjerknes' theory of 
pressure changes. That problem, as mentioned in section 10-06, deals 
with the study of the mechanism of pressure changes in the moving 
cyclones. In what follows we shall discuss the incipient wave in the 
baroclinic westerly current, characterized by the usual decrease of tem- 
perature along the isobaric layers towards the pole. To simplify the 
discussion we shall assume that the wave disturbance has infinite lateral 
extent with zero transversal divergence. We shall see later that the 
results are in principle the same for waves with finite width. 



Section 10-17 282 

The fields of longitudinal and transversal divergence have the zero 
lines along the troughs and along the wedges. Consequently, horizontal 
divergence does not explain the first formation and intensification of the 
wave disturbance. The only alternative is vertical motion, downward 
where the trough is formed and upward where the wedge is formed. 
This would explain why the wave pattern of upper isobars always forms 
with a definite phase lag, relative to the frontal wave disturbances in the 
lower atmosphere. The upward motion over the warm front surface 
creates the incipient upper wedge, and the downward motion behind the 
center at the ground creates the incipient upper trough. 

The complete description of that process in terms of the equations of 
dynamics and thermodynamics is not yet within reach, so we must con- 
fine our treatment to qualitative reasoning. Let us consider two limit- 
ing cases: 

1. The influence of the vertical displacement on the pressure change 
at the level is completely compensated by divergence or convergence 
in the air column above, so that everywhere ** 



This is the usual assumption when the vertical stability is examined. 

2. The influence of the vertical displacement on the pressure change 
at the level is not at all compensated by divergence or convergence in 
the air column above, so that everywhere 



The real case is likely to lie somewhere between (1) and (2) and will 
depend on the character of the initial undisturbed flow and the intensity 
and spatial extent of the vertical disturbance. 

Fig. 10-17a illustrates case (1). The upper part represents a vertical 
west-east cross section through the center of the young frontal wave. 
The intersection with the frontal surface appears in the diagram as an 
approximate sine curve tangent to the ground. In the reference level 
the isobars were initially straight and parallel to the path of the incipient 
wave, so that the line 00 on the vertical cross section was an isobar in the 
initial stage. According to equation (1) the isobar p = p Q stays in the 
position <t><t> also after the vertical motion has started. The isotherms 
in the reference level were initially straight and parallel to the isobars, 
so that 00 in the vertical cross section also was an initial isotherm. The 



283 . Section 10- 17 

vertical displacement A</> changes the temperature at the constant level <t> 
by 

(3) Ar= 



where yd is the dry adiabatic lapse rate, and 7 the actual lapse rate. 
The new isotherm position is consequently raised in the region of descent 
and lowered in the region of ascent. 



Front 



FIG. 10-17a. Vertical displacements with complete compensation by divergence in 

the air column above. 

In the initial stage the isobaric surfaces slope gently down toward the 
pole and the isothermal surfaces slope the same way at a steeper angle. 
The isobaric surfaces stay fixed also in the perturbed state. However, 
the isothermal surfaces move up in the region of descent and down in the 
region of ascent, thereby maintaining their meridional slope. Conse- 
quently, the horizontal map in the reference level </> (lower part of 
fig. 10-1 la) retains its straight west-east isobars also in the perturbed 
state. The isotherm T = 7"o, at first straight west-east and coinciding 
with the isobar p = po, becomes sinusoidal in the new state. Isotherms 
to the north and south of T = T behave similarly. They define a warm 
tongue extending northward in the region of descent, and cold tongue 
extending southwards in the region of ascent. 

The case (2) is illustrated in fig. l(M7fr. Since no horizontal diver- 
gence is present, no internal changes occur in the vertical air columns. 
They move up and down as solid columns. The vertical cross section 
(upper part of the diagram) shows how isobars and isotherms alike are 
lifted and lowered just as much as the air particles themselves. The 
lower part shows how sinusoidal isobars and isotherms result on the hori- 
zontal map in the reference level. The cold tongue coincides with the 



Section 10-17 



284 



pressure trough and the warm tongue with the pressure wedge. The 
north-south amplitude of the isobars is considerably greater than that of 
the isotherms, the reason being that the isobaric surfaces have a much 
smaller meridional slope than the isothermal surfaces. 

Figs. 10'17a and b show only the initial effect of the lowering and rais- 
ing of two adjacent portions of the baroclinic upper westerly current. 
After the upper wave has formed and the isobars and isotherms intersect 




FIG. 10-176. Vertical displacements without any compensation by divergence in 

the air column above. 

on the upper-level map, the horizontal advection starts to move the iso- 
therms. This process is shown in fig. 10-1 7c. The initial state has been 
selected as a compromise between the two limiting cases represented by 
figs. l(M7a and b. The initial isotherm in 10-1 7c has been made straight 
west-east, as a compromise between the sinusoidal isotherms with oppo- 
site phase in figs. 1017a and b. This initial straight isotherm is num- 
bered 0, and the subsequent positions reached by the same isotherm 
under the influence of horizontal advection relative to the wave are 
numbered 1, 2, , 6. We shall stipulate that the air moves eastward 
faster than the wave (the usual case in the upper air). The relative 
streamlines are then in phase with the isobars but have greater north- 
south amplitude (see fig. 10-155). The result of the advection is obvi- 
ously the formation of a warm tongue in the region of southerly wind 
components and a cold tongue in the region of northerly wind compo- 
nents. If the advection is allowed to continue undisturbed while the 
particles move one-quarter of the wave length, the warm tongue would 



285 



Section 10-17 



arrive at coincidence with the pressure wedge, and attain a north-south 
amplitude twice that of the relative streamlines. However, extrapola- 
tion of the isotherm advection so far ahead has little practical value 
because the isobars also change during the process. 




FIG. 10-1 7c. Horizontal advection of an initially straight isotherm. 

Let us consider an early stage in the isotherm advection in fig. 10-17c, 
and look for the change to be expected in the isobar pattern. The lower 





non- divergence 
Conv 



Div 





v -* 

Isobar 

Isotherm 



FIG. 10-17d. Structure of a young unstable wave in a baroclinic westerly current. 

part of fig. 10-17d shows the horizontal map at the level </>. The tem- 
perature and pressure waves are out of phase with each other, so the 
crests and troughs of the pressure wave must tilt, as shown in the upper 



Section 10-17 \ 286 

part of the figure. This upper diagram, representing the vertical cross 
section along the path of the wave, contains three successive isobars. 
The vertical distance between them is greatest in the warm tongue. 
The pressure crest therefore tilts westward (toward warm) with in- 
creasing height. And the pressure trough also tilts westward (toward 
cold) with height. 

The variation of the zonal wind with height in the warm air immedi- 
ately south of the polar front is shown in the right-hand diagram of 
fig. 10- 1 Td. The zonal wind increases with height in the troposphere to a 
maximum value at the tropopause and decreases from there up. The 
wave is assumed to move with a slightly slower speed than the warm air 
at the surface. This choice is plausible, since the surface air always over- 
takes the wave and ascends over the warm front. The relative zonal 
wind then increases from a small positive value at the ground to a maxi- 
mum value at the tropopause. Since the transversal divergence is zero, 
the flow is non-diverging at the level where the relative zonal wind is 
critical. The horizontal divergence ahead of the trough is positive 
above the level of non-divergence and negative below that level. In a 
westerly current of given strength the height of the level of non-diver- 
gence depends on the wave length. For short waves the level of non- 
divergence is low. As the wave length increases, the level of non- 
divergence is raised to greater heights until the tropopause is reached. 
For still longer waves no level of non-divergence exists. 

Let us first examine a short wave whose level of non-divergence is 
below the reference level </>, as indicated in the figure. Consider a verti- 
cal column of air extending upward from the reference level at the posi- 
tion of the trough line in that level. The pressure change at the base of 
this column indicates whether the trough is deepening or filling on the 
reference level map. First, the trough would deepen if the air continues 
to descend at the base of the column after the initial wave has developed. 
But, even if that downward motion has ceased, deepening would still 
continue if the integral of horizontal divergence in the column above the 
reference level shows depletion of air. Because of the westward tilt of 
the trough, the vertical column is east of the trough at all levels above the 
reference level. So the entire column has horizontal divergence, and 
the pressure tendency at its base is negative. Similar reasoning for the 
crest of the pressure profile shows that the pressure wedge on the map in 
the reference level must be building up, even though the upward motion 
which started the wedge may have ceased. Thus, a baroclinic westerly 
current is dynamically unstable for waves whose level of non-divergence is 
sufficiently low, and tends to make these waves grow strong, however weak 
they may be at the start. 



287 Section 10-17 

Let us next consider long waves where no level of non-divergence 
exists. The relative zonal wind at all levels is then subcritical, so the 
flow has horizontal convergence at all levels ahead of the trough. The 
long waves therefore would move westward. However, such waves 
would have no opportunity to develop. Since the trough and the wedge 
would tilt westward, the pressure would rise in the trough and fall on the 
wedge at all levels, and the initial disturbance would be damped out. 
Therefore the baroclinic westerly current is stable for these long waves. 

The waves which develop in the westerlies above the polar front 
always have finite lateral width and, consequently, have transversal 
divergence. The only way in which transversal divergence can influ- 
ence the above result is to make the total horizontal divergence zero for a 
somewhat smaller value of the relative wind than the critical speed. 
Other things being equal, the transversal divergence therefore lowers the 
level of non-divergence, and makes the waves of finite width more 
unstable than the infinitely wide waves. 

The same kind of dynamic instability exists whenever the temperature 
in a given isobaric surface decreases toward the left (right in the southern 
hemisphere) across the current, or, as it also may be expressed, whenever 
the speed of the current increases with height. The principal region of 
dynamical instability is thus the temperate region of westerlies. These 
westerlies are dynamically more unstable in winter than in summer. 
This explains, in a general way, the storminess in middle latitudes and 
its seasonal cycle. 

If the direction of the horizontal temperature gradient is reversed in 
the preceding discussion, we obtain a system where the troughs and 
wedges tilt eastward with height. Any disturbance in this system will 
therefore be damped out. In the stratosphere of the temperate zone 
the westerlies decrease with height. So independently intensifying 
waves seem excluded in this region. In winter at latitudes greater than 
60 the stratosphere temperature decreases toward the pole, so the 
westerlies must increase with height. In this region and season active 
wave formations may have their birth in the stratosphere and grow 
because of dynamic instability. 

Easterly currents are dynamically unstable only when they flow 
between a warm high and a colder low. This involves a reversal of the 
normal meridional temperature gradient, and is hence very rare in high 
latitudes. However, such a reversal is normal in summer in the low 
latitudes between the thermal and the geographical equators. So the 
equatorial easterlies in summer (especially late summer) exhibit some 
dynamical instability. This is probably significant for the formation of 
tropical cyclones. 



Section 10-18 288 

10*18. Horizontal divergence in closed cyclonic isobar patterns. The 

layers from the ground up to roughly 3 km elevation show more compli- 
cated pressure patterns than the higher layers. In the lower layers 
closed isobars around centers of high and low pressure frequently 
occur. Occasionally such closed pressure patterns also extend to layers 
well above 3 km. 

The simplest closed low (or high) for theoretical treatment is the one 
with the center at the pole, surrounded by circular zonal isobars coincid- 
ing with circles of latitude. Any ring-shaped channel between two 
neighboring isobars will then follow a constant latitude and will therefore 
have the same transport capacity throughout. So there is no region of 
accumulation or depletion of air and no pressure changes will occur. The 
circular vortex centered at the pole is thus a possible steady state. In 
fact it also exists in the atmosphere permanently. The upper westerlies 
of the temperate latitudes are part of the huge cyclonic vortex centered 
at the pole. (At the ground the polar low is concealed, since the cold 
polar air contributes enough surplus weight in the lower layers to make 
the minimum of pressure disappear.) 

If a low with concentric circular isobars is centered at some latitude 
away from the pole, the isobaric channels surrounding the center will not 
have equal transport capacity all around. Let us compare the transport 
capacity at the northernmost and southernmost points of the ring- 
shaped isobaric channel (fig. 10-18a). On account of the symmetry 
these points are places of maximum or minimum speed, where the wind is 
gradient wind. Assume for simplicity that the density is the same, and 
hence that the horizontal pressure force has the same value at the two 
points. When the normal component equation, 7-13(5), is differentiated 
with respect to latitude, keeping the curvature and the pressure force 
constant, we find 

bv vti cos <p 

MA __ _ . 

&<p Kv + 12 sin <p 

This shows that the gradient wind increases with decreasing latitude. 
The southernmost point of the ring-shaped isobaric channel has the 
maximum speed, and the northernmost point has the minimum speed. 
Since the isobaric channel has the same width at the two bends, the 
transport capacity is directly proportional to the wind speed. So the 
bottleneck of the channel is at the northernmost point. The whole 
western half of the vortex then exports more air to the eastern half than 
it receives in return. Air will be depleted from the western half of the 
vortex and accumulated in the eastern half. If the vortex extends with 
a vertical axis all the way up through the atmosphere, the pressure would 



289 



Section 10-18 



fall in its western part and rise in the eastern part. The low and the 
associated vortex would move westward. The stationary circular vortex 
reaching to the top of the atmosphere is thus impossible unless it is 
centered over the pole. 





FIG. 10-18a. Concentric circular iso- 
baric channel. 



<T 



FIG. 10-186. Eccentric circular iso- 
baric channel. 



The tendency for westward displacement of closed lows is counter- 
acted if the pressure pattern is made eccentric, as shown in fig. 1O18&. 
The isobaric channel is narrow in the south and wide in the north. 
According to equation 10-10(1) the transport capacity, dF=pv8ndz, 
will be equal at the southernmost and northernmost points if 

(2) Kv + 212 sin <p = K'V' + 2ft sin <p. 

If the central latitude y and angular amplitude <r p are introduced from 
10-10(4) and the subscript (p) is dropped, (2) becomes 

(3) Kv - K f v' = 412 cos sin cr. 

If the circular pressure pattern moves along the west-east direction, 
the relation between the curvature of the path and the curvature of the 
streamline at the southernmost point and the northernmost point of the 
isobaric channel is given by 

Kv = KS(V c), 
10-10(6) 

K'v' - K' s (v' + c). 

The circular isobar has the angular radius of curvature a and hence, in 
analogy to 7-07(4), its horizontal curvature is K p = I/ (a tan a). Since, 
now, the streamlines coincide very nearly with the isobars, we have 
approximately that K$ = K' s *= K p = I/ (a tan a), and consequently 

Kv - K'v' - K s (v - v' - 2c 



a tan <r 



When this expression is substituted in (3), we get the condition for equal 



Section 10-18 



290 



transport capacity at the southernmost and the northernmost points in a 
circular isobaric channel : 

(4) v v' 2c = 4 8 a cos <p sin a tan a. 

An isobaric pattern which in all isobaric channels satisfies equation (4) 
will be said to have the critical eccentricity, which makes the eastern and 
the western half of the vortex import equal amounts of air from each 
other during any given time. "We see from (4) that, approximately, 

v - v - 2c = 4 ft a<r 2 cos . 
The values of v v f 2c in (4) are given in table 10-18 for values of a 

TABLE 10-18 
(v t/ 2c) IN M s" 1 FOR CRITICAL ECCENTRICITY 



' 


<r 


1 


5 


10 


20 


90 














80 


0.1 


2.5 


9.9 





70 


0.2 


4.9 


19.4 


79 


60 


0.3 


7.1 


28.4 


116 


50 


0.4 


9.1 


36.6 


149 


40 


0.4 


10.9 


43.6 


177 


30 


0.5 


12.3 


49.3 


200 


20 


0.5 


13.3 


53.5 


217 


10 


0.6 


13.9 


56.0 






up to 20. The table shows what velocity difference there must be 
between the southernmost and the northernmost point of a stationary 
ring-shaped isobaric channel in order that the channel shall have equal 
transport capacities at the two points. The value of v v f 2c for the 
critical eccentricity increases with increasing distance from the pole 
and increasing radius of the channel. A stationary low with exactly the 
critical eccentricity for all isobaric channels would have almost concen- 
tric isobars near the center and increasingly eccentric isobars toward the 
outskirts. Isobaric channels of 20 radius require impossible winds to 
have the critical eccentricity. 

If the low moves toward the east, the values in the table must be aug- 
mented by twice the speed of the low to give the velocity difference v - v' 
needed for critical eccentricity. If the low moves westward the tabu- 
lated values must be diminished by the same amount. So the critical 
eccentricity is accentuated if the low moves toward the east and less 
pronounced if the low moves toward the west. 



291 Section 10-19 

The actual eccentricity of closed flow patterns on the map, as 
expressed by the difference in wind velocity at the southernmost and 
northernmost points of individual isobaric channels, may be compared 
with the tabulated values for the critical eccentricity. It will generally 
be found that closed circulations have subcritical eccentricity. That is, 
they do not have enough eccentricity to make the eastern half of the 
vortex export as much air to the western half as it receives in return. 

The total accumulation or depletion of air in a half-vortex depends not 
only on the exchange of air with the other half of the vortex but also on 
the inflow or outflow across the limiting outer isobar. In an eccentric 
vortex, like that in fig. 10-18&, the flow of air from the velocity maximum 
in the south to the velocity minimum in the north would be associated 
with a slight component of motion across the isobar towards high pres- 
sure. This represents a depletion of air from the eastern half of the 
vortex. Corresponding reasoning for the western half shows an inflow 
across the outer isobar and accumulation of air in that part. 

Thus eccentricity tends in two ways to deplete air from the eastern 
half of the vortex, and accumulate air in the western half, and thereby 
counteracts the latitude effect which makes the eastern half gain and the 
western half lose air. The flow is actually non-diverging for slightly 
subcritical eccentricity. 

10*19. Closed cyclonic isobar patterns surmounted by wave-shaped 
patterns. The above treatment of the cyclonic vortices in temperate 
latitude tends to show that only those low-pressure patterns can move 
eastward which have packed isobars to the south and open isobars to the 
north of the center. The lower the latitude the greater the eccentricity 
required. However, that result is valid only for cyclones which reach 
with essentially the same closed isobaric patterns to the top of the 
atmosphere. The usual cyclone of the temperate latitudes has closed 
isobars only in the lowest 2 or 3 km of the atmosphere. Higher up it 
appears only as a trough in the upper west-east trend of the isobars. 
The treatment of such cyclones will show that, although a pattern of 
packed isobars to the south and open isobars to the north of the center is 
favorable for a rapid displacement eastward, such a pressure pattern is 
by no means necessary for the eastward displacement of the ordinary 
cyclone in temperate latitudes. 

Fig. 10' 19 represents a schematic picture of the processes which 
produce the pressure changes in the ordinary eastward moving 
cyclone in temperate latitudes. It has been assumed that (v - t/) < 
412 acr 2 cos + 2c. According to 10*18(4) this implies that the pressure 
distribution around the closed center in the lower atmosphere is not 



Section 10-19 



292 



sufficiently eccentric to cause a depletion of air in front of the center, and 
an accumulation of air behind. In the layers with closed isobars we have 
horizontal convergence in front of the cyclone and horizontal divergence 
in the rear. Higher up, where the cyclone is represented by a trough in 
the west-east isobars, there will, according to our earlier results, be hori- 
zontal divergence in front of the trough and horizontal convergence 
behind. 



Upper level 
isobars 

Surface 
isobars 




Sinusoidal 
isobars 



Closed isobars 
Friction layer 



Propagation 

FIG. 10-19. Closed cyclonic isobar pattern surmounted by wave-shaped pattern. 

A column fixed in space at A (while the air is flowing through it) will 
gain weight by the convergence in the lower layers and will lose weight 
by the divergence in the upper layers. In order that there shall be a net 
loss of weight of the whole column, more air must be removed in the 
upper layers than accumulates through convergence in the lower layers. 
The fall of pressure ahead of the cyclone depends on that. 

A column fixed in space at B will gain some weight by the convergence 
in the friction layer, but will normally lose more than that by the diver- 
gence in the superjacent layers of closed isobars. Above the level where 
the closed center disappears, column B will gain weight by the conver- 



293 Section 10-19 

gence prevailing behind the upper air trough. The net effect of the 
changes in weight of column B should be positive, so as to give a rise in 
pressure behind the moving cyclone. Again this demands that the 
influence from the upper layers shall overcompensate the influence from 
the lower layers. 

Fig. 10-19 explains in a qualitative way how centers of low can have 
falling pressure in front and rising pressure in the rear, although the 
analysis of the horizontal flow in the low layers shows accumulation of 
air in front of the center and depletion of air in the rear. The general 
west-east drift of centers of low is thereby explained very much in accord- 
ance with the old rule of thumb : The centers move along with the upper 
current. We now have obtained the physical explanation for this rule: 
The upper current provides for depletion of air in front of the center and 
accumulation of air behind the center. 

Actually there is a phase difference between the upper trough and the 
cyclone center at the ground, the upper trough lagging a little. Because 
of that phase lag upper air divergence takes place vertically above the 
central area of the cyclone. If the upper air divergence at that place is 
strong enough to overcompensate the convergence of air into the cyclone 
center in the lower layers, there will be falling pressure at and around the 
center. In that case the cyclone is deepening. This process of intensi- 
fication evidently depends on the phase lag and the backward tilt of the 
upper trough, which again (according to section 10*17) reverts to the 
vertical shear as the primary cause for wave instability. 

It is furthermore evident from fig. 10-19 that deepening is most likely 
to occur if the pressure pattern changes from closed isobars to wave- 
shaped isobars at a relatively low level. This is characteristic of young 
cyclones. The first level to have closed isobars around a new cyclone is 
the surface level. As the cyclone grows it develops closed isobars at 
successively higher levels. The older the cyclone, the higher up will be 
the level of transition from closed to wave-shaped isobars, and the more 
the pressure changes will be influenced by the lower pattern of closed 
isobars. Finally the influence of the lower pattern will cancel or over- 
compensate that of the upper pattern, and the cyclone will stop or turn 
slightly retrograde. This, too, is very well corroborated by nature. 
Cyclones move fast eastward while they are young, but slow down when 
they get old and deep, sometimes even retrograding a little toward the 
west in their last phase. 

It is interesting to note that the vortex in the surface layers, while 
being forced along by the influence of the " upper current," counteracts 
that displacement by piling up air in front. Thus if a vortex of given 
eccentricity is forced to move to the east, 'that eccentricity becomes the 



Section 10-19 294 

more subcritical, the faster its eastward speed. In an eastward moving 
low the air will accumulate in the eastern half at the expense of the air 
in the western half, even though the same pressure pattern when station- 
ary maintained non-divergence. This again means that the pressure fall 
in the front half and the rise in the rear half, both originating in the 
upper current, will be reduced respectively by the convergence and 
divergence in the lower layers. This effect increases with increasing 
eastward speed of the system and thus acts as a regulator of the speed. 

The next step in the treatment of the pressure changes in moving 
cyclones should be to modify the circular isobars at the surface so as to 
accommodate fronts. Obviously there are a great many problems left 
for future work concerning the distribution of divergence in pressure 
fields with non-circular isobars delimiting more or less sharply defined 
troughs, etc. However, it is expected that the fundamental difference 
between the front and the rear of the moving cyclone, as represented in 
fig. 1(M9, will remain as a background pattern upon which frontal effects 
are superimposed. 

1020. Closed anticyclonic isobar patterns. The analysis of pressure 
changes associated with moving highs can be carried out in analogy with 
the above analysis of the pressure changes in moving lows. 

The anticyclone with concentric circular isobars is impossible as a 
steady state except when centered at the pole. At other latitudes 
steady-state anticyclones extending to high levels must have eccentric 
isobars, with the maximum pressure gradient on the polar side of the 
center. The usual moving anticyclone does not extend far from the 
ground as a system of closed isobars, but it is surmounted by a wedge. 
The upper wave-shaped flow pattern produces accumulation of air over 
the front half of the high and depletion of air over the rear half, and 
thereby makes it move eastward. The lower closed flow pattern pro- 
duces the opposite effect, but it is overcompensated by the contribution 
of pressure change superimposed from the high layers. Since the upper 
wedge usually lags behind the position of the center of high at the 
ground, the upper flow pattern produces an accumulation of air over the 
central region of the high. If that accumulation of air is sufficient to 
overcompensate the outflow of air in the frictional layer near the ground, 
the high will intensify. 



CHAPTER ELEVEN 
CIRCULATION AND VORTICITY 

11*01. Method of line integrals. Many problems of atmospheric 
motion are studied quite conveniently by a special method which 
involves the application of line integrals. This method has a close 
analogy to the general method used in the mechanics of rigid bodies. In 
this field of science all internal deformations are neglected, and the body 
is studied as a rigid entity. The internal forces in the body, which 
appear in pairs, are thus eliminated from the problem. 

Similarly, in hydrodynamics we may consider as an entity a group of 
fluid particles lying on a closed curve, and then neglect all differences 
between the particles in the group. The dynamics of such a group is 
determined by equations with one of the acting forces eliminated. 
These equations are much simpler than the general equation of motion. 

11 -02. Line integral of a vector. Each of the physical vectors, 
velocity, acceleration, and gravity, is defined at every point in the 
atmosphere. Let a denote any one of these vectors. In the field of a 
consider an arbitrary curve connecting an initial point with the position 
vector TI and a terminal point with the position vector r 2 , fig. 11 *02. 
Let this curve be divided into infinitesimal curve elements. Each curve 
element defines an infinitesimal 
vector 5r, whose sense is given 
by the direction from rj to r 2 
along the curve. Since the curve 
is located in the field of a, the 
vector a has at any given time a 

definite value at each point on n ^^ ] a r 

the curve. When scalar multi- 
plication of each of the elements 
5r by the corresponding vector a is performed, we obtain infinitesimal 
scalar products. The sum of these products defines by its limit the 
curvilinear integral (line integral) of the vector a, which is denoted as 

follows: 

2 




(1) 



' = / aSr. 



i 
295 



Section 11-02 296 

Mathematically there is no difference between this integral and those 
defined in ordinary calculus. Both are limits of sums of infinitesimal 
quantities. In (1) the integral element is the scalar product which, 
according to 4'07, may be written in any of the following scalar forms: 

(2) a-5r = arfs = ads cos 6 = a x dx -h a y dy -f a z f>z. 

We shall, however, prefer to use the compact expression (1). 

It follows from (2) that only the tangential component a^ of the vector 
is subjected to the integration. In the special case where the vector is 
the velocity, the integral (1) is known as the procession and is denoted 
byP; thus 



/ 



A similar terminology is used in the general case (1) where the physical 
nature of the vector is unspecified, and the integral (1) is called the pro- 
cession integral of the vector a. 

When the curve of integration in (1) is a closed curve, the initial and 
the terminal points will coincide, and the integral becomes a cyclic 
integral. Denoting the closed curve by c we shall introduce for 
the curvilinear integral of a around the closed curve the notation 



(3) 



/a'Sr. 



In the special case when the vector is the velocity, this integral is called 
the circulation, and is denoted by C; thus 



(3') C= / 

J c 



In the general case (3) the integral is called the circulation integral 
of the vector a. 

It is important to keep in mind the complete generality of the integral 
defined above. The geometrical properties and the physical nature of 
the vector field may be completely arbitrary, and the curve of integra- 
tion may have any shape and location in the field. Thus there is a 
priori no relation between the vector field and the curve, except that the 
entire curve must be within the region where the vector is defined. The 
physical interpretation of the integrals is evident in the special case 
where the vector is the velocity. The interpretations for other physical 
vectors are given in the following sections. The evaluation of the 
integrals can only be performed when the vector field and the curve of 



297 Section 11 -03 

integration are known. However, our aim is not to evaluate the inte- 
grals, but rather to develop general laws for their behavior. These laws 
make it possible to investigate further the dynamical relations between 
the motion and the fields of the physical variables. 

11*03. Line integrals of the equation of absolute motion. We shall 
first consider absolute motion, where a simple physical interpretation 
of the results is possible. The equation of absolute motion can be 
written in the following two equivalent forms: 



PV - - 

where in (la) the equation is referred to unit mass, and in (16) to unit 
volume. 

Following the procedure outlined in the preceding section, we perform 
scalar multiplication of each of the vectors in these equations by the 
vector line element 5r of an arbitrary curve in the atmosphere. Accord- 
ing to 4-13(1) the scalar products of the potential vectors are 



(2) V<t> a *dr = 50 a , V/>-5r = dp; 

here 6< a and dp are the variations respectively in the potential of gravi- 
tation and in pressure from the initial to the terminal point of the vector 
element dr. Thus the result of the scalar multiplication in (1) is 

(36) pv a 5r = pd<t> a dp. 

Each of the expressions may be integrated along the curve from an 
initial point 1 to a terminal point 2 on the curve. We thus obtain the 
following relations between the procession integrals of the vectors in the 
equation of motion : 

2 2 

(4a) <t>az - i - - ] adp - J v a .5r, 

1 i 

2 2 



(46) 



P2 - Pi - ~ / P$*a - / 



The first of these equations is the dynamic generalization of the baro- 
metric height formula. 

Each of the equations (4) remains a full equivalent of the equation of 
motion, as long as we have free choice of the curves. This generality is 



Section 11-03 298 

lost when the curves are subjected to special conditions, but in return we 
obtain useful special theorems. We shall consider two such speciali- 
zations : 

1. Equilibrium curves. In any field of motion an infinite number of 
curves may be drawn normal to the acceleration. These curves are 
called equilibrium curves. If the integrals (4) are taken along an equi- 
librium curve, the equations reduce to the hydrostatic forms: 



(5a) 



z 
! ~ <t>al = - / Otdp, 



(56) P2 - Pi 



X 



Thus the barometric height formula, which is approximately fulfilled 
for an arbitrary curve, is fulfilled exactly along the equilibrium curves. 
These curves may, under special simple conditions, be rather easy to 
determine. For example, for steady zonal motion the only acceleration 
is the centripetal acceleration, so that one system of equilibrium curves 
consists of the lines parallel to the axis of the earth ; another system 
consists of the circles of latitude. 

2. Isobaric curves and horizontal curves. If the curve in ^4a) lies in an 
isobaric surface, and the curve in (46) in a level surface of <f> a , the equa- 
tions reduce to 



z 

- / v a -5r, 



2 



(66) p 2 - Pi 



- /pv a -5r. 



The first of these formulas indicates that the isolines of <t> a on an isobaric 
surface run normal to the isobaric component of the acceleration; the 
other, that the isobars in a level surface of <t> a run normal to the compo- 
nent of the acceleration in that surface. 

11*04. Primitive circulation theorems in absolute motion. We 

shall next consider the case where the line integrals in the two equations 
1 1 '03 (4) are taken along a closed curve. The initial and terminal points 
of the integration are then identical, and the procession integrals become 



299 Section 11-04 

circulation integrals; thus: 
(la) (v -5r=- I adp, 

J c J c 



/ pv a -5r = - / pS<f> a . 

Jc J c 



These equations connect the circulation of the vectors v a and pv a with 
cyclic integrals of the physical variables. In (la) the potential of gravi- 
tation is eliminated, and in (16) the pressure is eliminated. They are 
known as the circulation theorems and were derived by V. Bjerknes in 
1898. Both theorems are fundamental for the study of physical hydro- 
dynamics. 

The theorem (16) has been given a useful mechanical interpretation 
by E. Hoiland (1939). He considers a closed fluid tube with the 
infinitesimal constant cross section 5 A and applies the theorem (16) to a 
curve which is the central line in this tube. If we multiply (16) by the 
constant 5 A and introduce -5$ = -V$ a *5r = g a *6r, we have: 



/ 8Apv a -8r = I 

J c J c 



(2) / &4pv -5r= / 8Apg a 9 8r. 

J c J c 

Let VT and g a r denote the components of the acceleration and the gravi- 
tation tangential to the tube in the direction of the vector element 5r. 
With these notations equation (2) takes the form: 

(2') I v r (p8A8s) - J g aT ( P 8A8s), 

or, since p8A8s - 8M is the mass contained in the tubular volume 
element 8V = 8 A 8s: 



(2") I v T 8M= I 

Jc Jc 



This equation, which is equivalent to the circulation theorem 
clearly reveals the analogy between this theorem and the equation of 
motion of a rigid body. By considering the particles in a closed tubular 
fluid filament we have a mechanica system for which the pressure forces 
have no resultant in the direction tangential to the filament. The inte- 
gral of the mass multiplied by the acceleration tangential to the fila- 
ment is therefore determined by the gravitation alone. Hoiland calls 
the integral f c i>T8M the total mass acceleration along the filament. In 
this terminology the circulation theorem (16) can be stated as follows: 
An arbitrary closed fluid filament with constant cross section has a total mass 



Section 11-04 300 

acceleration along itself equal to the resultant of the force of gravitation along 
the filament. This theorem is particularly important in the study of the 
stability of steady flow. 

The theorem (la) can be given a similar mechanical interpretation by 
considering a closed fluid filament of cross section dA , -where pdA is a 
constant along the filament. The particles composing this fluid fila- 
ment represent a mechanical system where the resultant of gravitation 
tangential to the filament is zero. Another interpretation of the circu- 
lation theorem (la), which has been of great importance for the develop- 
ment of modern meteorology, has been given by V. Bjerknes. This 
interpretation will be discussed in the following section. 

, ," 11 -05. The theorem of solenoids. It was shown in section 423 that 
the pressure integral fadp has a simple'graphical representation in the 
(c*,-/0-diagram. Corresponding values of pressure and specific volume 
along a vertical curve in the atmosphere define a certain curve in the 
(a, -p) -diagram, or in the emagram, as in fig. 4-23. This curve, known 
as the sounding curve, represents the vertical curve in the atmosphere, 
and shows how the fields of pressure and mass are distributed along this 
vertical. 

Consider now an arbitrary geometrical curve in the atmosphere. At 
any given instant this curve may be represented in the (a, p)-diagram 
by a curve which is determined by the distribution of pressure and mass 
along the atmospheric curve. We shall refer to the latter as the image 
curve of the atmospheric curve. When the atmospheric curve is closed, 
its image curve is also closed. The pressure integral f c adp in the cir- 
culation theorem ll-04(la) is equal to the area enclosed by the image 
curve, and its sign is determined by the sense of the integration around 
this curve. The sign is positive if the integration has the sense of the 
rotation from the positive a axis to the negative p axis, and negative if 
the integration has the opposite sense. 

Theisobaric unit layers in the atmosphere are represented in the (a?, /?)- 
diagram by horizontal stripes of unit width, and the isosteric unit layers 
are represented by vertical stripes of unit width. Taken both together 
the isobaric and isosteric unit surfaces divide the atmosphere into a 
system of tubes with parallelogrammatic cross sections. These unit 
tubes are known as the pressure-volume solenoids or the (a,-)-sole- 
noids. Each solenoid in the atmosphere is represented by a unit square 
in the (a, -p) -diagram, as shown in fig. ll05a. The sign of the unit 
square is positive when the integration along its edge has the sense of the 
rotation from the positive a axis to the negative p axis. Correspondingly 
a positive sense may be assigned to the solenoid, defined by the right- 



301 



Section 11-05 



handed rotation (through an angle less than 180) from the volume 
ascendent Va to the pressure gradient -Vp. (In fig. ll05a the sole- 
noids are directed out from the paper.) 

Consider an arbitrary closed curve in the atmosphere and the corre- 
sponding image curve in the (,-/>) -diagram. The two curves are 
denoted respectively by c and c f in fig. 1 1 05a. The area enclosed by the 




o* * 






t 

o o 

+ + 

CO * 



FIG. ll-05a. Pressure- volume solenoids enclosed by atmospheric curve, and equiv- 
alent area on (a, )-diagram. 

image curve c 1 equals the number of unit squares contained within the 
curve. And since the integral -f c a5p is equal to this area, it is also 
equal to the number of pressure-volume solenoids embraced by the 
atmospheric curve c. The solenoids are counted algebraically. If their 
sense, as defined above, is the same as the sense of integration along the 
curve, they are counted positive; if their sense is opposite, they are 
counted negative. In fig. ll-05a the sense of integration indicated by 
the arrow on the curve is the sense of the solenoids. The solenoids are in 
this case counted positive, and the integral -f c a5p, taken in the indi- 
cated sense, is accordingly positive. Had the integration been taken in 
the opposite sense, the solenoids would have been counted negative. 
Denoting the algebraic number of solenoids embraced by an arbitrary 
closed curve in the atmosphere by N a ^ p , we can write 



(i) 



- I a8p 

J c 



This is the theorem of solenoids. When this theorem is combined with 
the circulation theorem ll-04(la), we obtain 



(2) 



N< 



a, p- 



In this form the circulation theorem can be stated: Along an arbitrary 
closed curve in the atmosphere the absolute acceleration has at any time a 
circulation equal to the algebraic number of pressure-volume solenoids 



Section 11-05 



302 



embraced by the curve. The circulation has the same sense as the solenoids, 
the sense of the rotation from volume ascendent to pressure gradient. 

This interpretation of the circulation theorem was given by 
V. Bjerknes. It does not reveal the underlying mechanical principle as 
clearly as Holland's interpretation. But the solenoid theorem has great 
practical advantages, particularly in the science of synoptic meteorology, 
where the physical fields are represented graphically by their unit layers, 
so the solenoids are directly accessible. 

As an example consider the ideal case shown in fig. ll-OSfe. The dia- 
gram to the left represents a meridional cross section through the lower 
troposphere, symmetric with respect to the axis of the earth. It shows 
schematically the mean distribution of pressure and mass in winter. 
The isobaric surfaces are drawn for every 10 cb, the isosteric surfaces for 
every 100 m 3 F" 1 . The mts-solenoids are thus obtained by increasing the 
number of isobars ten times, and the number of isosteres one hundred 
times. The solenoids are in this case annular tubes parallel to the circles 
of latitude, and their sense is from east to west. Consider the closed 
atmospheric curve 1231, composed of the isobaric segment 12, the iso- 
steric segment 23, and the vertical segment 31 along the axis of the earth. 



Vcr 



-Vp 





700 800 900 

Of*- 

FIG. ll-OSb. Idealized distribution of solenoids in meridional cross section. 

The corresponding image curve in the (<*,-)-diagram, to the right in 
fig. 11*056, is the triangle enclosing the shaded area. We shall now 
apply the circulation theorem (2) to this curve, integrating along the 
curve with the sense indicated by the numbers. The area of the image 
curve is then positive, and the solenoids embraced by the atmospheric 
curve have the same sense as the integration, the sense of the rotation 
from volume ascendent to pressure gradient. Both results show that the 
acceleration has a circulation along the curve in the direction indicated 
by the numbers. The same result is obtained when the circulation 
theorem is applied to the curve 1431. 



303 Section 11-06 

Consider finally the atmospheric curve 12341, whose image curve is 
the broken line 12321, which has zero area. The acceleration has no 
circulation along this curve. This result is also obtained directly from 
the theorem of solenoids. The solenoids to the right of the axis have the 
same sense as the integration and are counted positive. The solenoids 
to the left of the axis have opposite sense and are counted negative. 
Due to the symmetry there are equal numbers of positive and negative 
solenoids, so the total algebraic number of solenoids embraced by the 
curve 12341 is zero. 

The above example shows that the information which can be gained 
from the theorem is greatly influenced by the selection of the curve. The 
curves 1231 or 1431 give the useful information that the acceleration has 
a tangential resultant along these curves in the direction indicated by 
the numbers. The curve 12341 gives the trivial result of no tangential 
resultant. One guiding principle for the selection of the curve is that it 
should embrace only solenoids of the same sign. Only with such curves 
can the full dynamical effect of the solenoids be estimated. 

11-06. Practical forms of the theorem of solenoids. Since pressure 
and specific volume are the physical variables which enter directly into 
the atmospheric equations, the pressure-volume solenoids give the 
simplest rules when applied to dynamical problems. This is analogous 
to the fact that the (a,-/?) -diagram is the simplest therrnodynamical 
diagram for the study of thermal energy transformations. 

In practical meteorology the (,/>) -diagram is replaced by the ema- 
gram or the tephigram, which have the important meteorological vari- 
ables as their coordinates. Similarly, the pressure-volume solenoids 
may be replaced by the pressure-temperature solenoids or the tempera- 
ture-entropy solenoids. The relations between these three kinds of 
solenoids are the same as the relations between the (a,-p)-diagram, the 
emagram, and the tephigram. If the same closed atmospheric curve is 
plotted in these three diagrams, the areas enclosed by the corresponding 
image curves in the diagrams are proportional; thus 

(1) - / abp = R d / In p dT - -c pd / In 8T. 

Jc Jc Jc 

The first integral is the pressure integral in the circulation theorem and 
is equal to the number of (a, -p) -solenoids embraced by the atmospheric 
curve. The second and third integrals in (1) may be given a similar 
interpretation. 

The second integral is equal to the area enclosed by the image curve 
in the emagram. It is positive if the integration has the sense of the 



Section 11-06 304 

rotation from the negative In p axis to the negative T axis. This area is 
also equal to the number of ( In /?, 7") -solenoids defined by the iso- 
thermal unit layers and the unit layers of In p. The sense of these sole- 
noids is defined by the rotation V- V7" from pressure gradient to 
temperature gradient. The number of these solenoids embraced by the 
atmospheric curve will be denoted by N\ n pt _ T . 

The third integral is equal to the area enclosed by the image curve in 
the tephigram and is therefore also equal to the number of (In 0, T)- 
solenoids. These solenoids are defined by the isentropic and the iso- 
thermal unit layers and have the sense of the rotation V(ln 0) V7' 
from entropy ascendent to temperature gradient. The number of these 
solenoids which are embraced by the atmospheric curve will be denoted 
by Ni net ^ T . Introducing these notations in (1), we obtain 

(2) N at _p = RdN^i n pt ^ T = CpdN\ n Q ,-T* 

The three kinds of solenoids in (2) are always parallel. The pressure- 
volume solenoid is by definition parallel to the line of intersection of the 
isobaric and the isosteric surface, along which both pressure and specific 
volume are constant. From the equation of state the temperature is 
constant along this line, and from Poisson's equation the potential tem- 
perature is also constant along the same line. 

In synoptic analysis the isobars and isentropes are drawn for unit 
values of p and 0. The determination of the solenoids in (2) requires the 
isobars and isentropes for unit values of In p and In 0. To overcome this 
practical inconvenience, consider one of the temperature-entropy sole- 
noids, defined by two isothermal surfaces with the temperature differ- 
ence A T - 1, and two isentropic surfaces with the entropy difference 
A(ln0) = 1. The variation A0 in potential temperature through the 
isentropic unit layer is given approximately by A0/0 = A (In 0) = 1, or 
A0 = 9, where S is a mean potential temperature in the layer. Accord- 
ingly, an isentropic unit layer contains potential temperature unit 
layers, and one temperature-entropy solenoid contains temperature- 
potential temperature solenoids. Denoting by Ne,-r the number of the 
latter solenoids embraced by the atmospheric curve, we then have 

(3) N0 t -T - ^in^.-r. N-p,-:r = />AT-inp,-r- 

The second expression is obtained by a similar consideration of the 
( -In />, r)-solenoids. When the expressions (3) are introduced in (2), 
we have 



305 Section 11-06 

This gives the relations between the number of pressure-volume sole- 
noids, pressure-temperature solenoids, and temperature-potential tem- 
perature solenoids which are embraced by the same atmospheric curve. 
In a dry atmosphere the solenoids are defined by the real temperature 
and potential temperature. In a moist atmosphere the virtual tempera- 
ture and virtual potential temperature must be used; see 3-28. 

Practical forms of the circulation theorem are obtained when the 
second and third expressions in (4) are introduced in the fundamental 
theorem 11-05(2); thus 



(5) -r- 

Jc Pi 6 

When the unit isotherms, and the unit isobars or the unit potential iso- 
therms have been drawn in a vertical cross section through the atmos- 
phere, one of these theorems may be used to obtain a general idea of the 
distribution of the acceleration in the cross section. For rough esti- 
mates, the numerical factors (Rd/p) and (c p d/~) are obtained from 
mean values of p or 6 within the curve. The variation in the factor 
(cpd/B) is small within the lower half of the troposphere where the range 
of potential temperature is about 270-330. A rough average value of 
(cpd/0) is thus 3.3, which gives the approximate rule: Each tempera- 
ture-potential temperature solenoid contains about 3 pressure- volume 
solenoids. 

The formulas (5) are used only for qualitative estimates of the dis- 
tribution of the acceleration when the physical variables are represented 
graphically by their unit layers. When the solenoid number is wanted 
more accurately, it is determined by integration around a " rectangu- 
lar " curve composed of two isobaric and two vertical curve segments. 
According to (1) we have for any closed atmospheric curve 



-Rd I 

Jc 



(6) N at - p =-R d I T*(lnp). 

Jc 

The two isobaric curve segments give no contribution to the integral on 
the right. Along .the vertical segments of the curve the integral is 
identical to the barometric height formula. Let pi and p 2 be the pres- 
sures respectively on the lower and the upper isobar, and let A and B 
denote the colder and the warmer vertical, having respectively the mean 
temperatures T mA and T mR . Denoting by <t> A and <f> B the dynamic 
thicknesses of the isobaric layer (pi - p 2 ) at tne two verticals A and B, 
we see from 4-24(3) that 

(7) N a ,- p \ = R d (T mB - T mA ) In ^ - ^ - + A . 

P2 



Section 11-06 



306 



With the aid of the second expression the number of solenoids is con- 
veniently evaluated. The problem has been reduced to the determina- 
tion of the dynamic heights of the two verticals. The last expression in 
(7) shows that the number of solenoids contained in the isobaric layer 
between the two verticals is equal to the variation in dynamic thickness 
of the layer, or the dynamic inflation of the isobaric layer. In the special 
case of an isobaric unit layer this result is obtained directly from 4-16(5), 
which contains the rule that the dynamic thickness of the isobaric unit 
layer is equal to a. The dynamic inflation of the isobaric unit layer 
from A to B is a H <X A , which evidently is equal to the number of sole- 
noids in the layer between A and B. 
Table 11-06 has been computed from the formula (7), with the pressure 

TABLE 11-06 
NUMBER OF PRESSURE- VOLUME SOLENOIDS 



^ 


T mB - T mA 


1 


10 


20 


30 


40 


30 


346 


3,455 


6,910 


10,365 


13,820 


60 


147 


1,466 


2,932 


4,398 


5,864 


90 


30 


303 


605 


908 


1,210 


100 


















on the lower isobar at the standard value pi = 100 cb. The table gives 
the number of solenoids contained in a curve bounded by this lower 
isobar, an upper isobar of pressure p 2 and two verticals with the mean 
temperature difference T mB T mA . 

As an example consider a closed meridional curve having one vertical 
at the pole, the other at the equator, and having the upper isobar 
p2 SB 30 cb and the lower isobar p\ 100 cb. This curve will embrace 
the majority of the solenoids of the troposphere. In winter the difference 
of mean temperature between the two verticals is about 40. It is seen 
from the table that the corresponding number of solenoids is 13,820. 
This result also signifies that the isobaric layer 100-30 cb has a dynamic 
inflation of 1382 dyn m from the pole to the equator. 

11-07. Dynamic balance of steady zonal motion. The utility of the 
circulation theorem 1 1-05 (2) for the study of atmospheric motion may be 
illustrated by considering steady zonal motion. This field of motion is 
sometimes referred to as a circular vortex. The absolute acceleration is 
here the centripetal acceleration -co^R. For an arbitrary curve in the 



307 



Section 11-07 



circular vortex the circulation theorem becomes 

a) 



- f <iL-dT = N a ._ p . 

J c 



We shall first examine the distribution of the solenoids in the circular 
vortex. It was shown in section 6-11 that the pressure field is symmetric 
about the axis of the earth. It is easy to see that the mass field is also 
symmetric about the axis. Since the isobaric surfaces are surfaces of 
revolution, the pressure gradient -V has a constant magnitude along 
any one of the circles of latitude. According to the equation of motion 
the pressure force per unit mass -aVp is constant in magnitude along 
each latitude circle. The specific volume is the ratio of the magnitudes 
of the pressure force and the pressure gradient. Hence, a = |V/>|/|V/>| 
is constant along each latitude circle, which proves that the isosteric 
surfaces are surfaces.of revolution about the axis of the earth. The sole- 
noids are therefore annular tubes parallel to the circles of latitude. 



-Vp 





3 



"S 

(/) 

3 



R 




FIG. ll'07a. Solenoids in balanced 
zonal motion. 



FIG. 11-076. Circulation theorem ap- 
plied to rectangular meridional curve. 



The sense of the solenoids is determined by the rotation from volume 
ascendent to pressure gradient. Under normal conditions the cold and 
heavy masses are located in the polar region, as shown in fig. ll07a. 
In this normal atmospheric vortex with a cold core the solenoids are 
in the northern hemisphere directed from east to west. In the abnormal 
vortex with a warm core the solenoids have the opposite sense. 

From the circulation theorem (1) the tangential resultant of the 
centripetal acceleration along any closed meridional curve has the same 
sense as the solenoids, Consider first the normal vortex with a cold 
core. Let the meridional curve be a rectangle, with sides parallel and 



Section 11-07 308 

perpendicular to the axis, fig. 11*07&. The " height " of the rectangle 
in the direction of the axis is arbitrary, but its width normal to the axis 
is infinitesimal, denoted by 61?. Let further co a2 and co i denote the 
angular speed at the top and the base of the rectangular curve. 
(Here " up " and " down " refer to the equatorial plane.) Since the 
axial sides of the rectangle are equilibrium lines (perpendicular to the 
acceleration), they give no contribution to the circulation integral in 
(1). Thus, when the integration has the sense of the solenoids, 



(2) (u 2 a2 - 

The solenoid number is positive, so u? l2 > ufai* The rectangular curve 
maybe placed anywhere in the meridional plane ; therefore throughout 
the vortex the angular speed increases with increasing distance from the 
equatorial plane. When a similar analysis is applied to the abnormal 
vortex with a warm core, we find that the angular speed decreases with 
increasing distance from the equatorial plane. When the specific volume 
is constant within each isobaric layer, the vortex contains no solenoids 
and is said to be barotropic. The angular speed is constant along any 
line parallel to the axis in the barotropic vortex. When " height " 
refers to the equatorial plane, the above results may be condensed into 
the following rule . The strength of rotation in the circular vortex increases 
with height when its core is cold, decreases with height when its core is warm, 
and the rotation is independent of height when the vortex is barotropic. 

It is possible to derive these rules directly from simple physical reason- 
ing. When the circulation integral in (1) is transferred to the right-hand 
side, 



(3) 



0-/4 

Jc 



the theorem becomes an equation of equilibrium. The two terms on the 
right must be balanced for any meridional curve if the vortex shall be 
maintained as a steady state. Consider the dynamical effect of each 
of the terms in the normal atmospheric vortex with the cold core, as 
illustrated in fig. ll'07c. The solenoids tend to produce a " direct " 
meridional circulation around the solenoids: the heavier masses in the 
polar region tend to sink down and spread out southward along the sur- 
face of the earth, with a compensating northward flow aloft of the lighter 
masses from the equator. This circulation is prevented by the circula- 
tion of the centrifugal force. The strength of the zonal circulation and 
hence the centrifugal force increases with " height,' 1 so the centrifugal 
forces tend to produce a " retrograde " circulation around the sole- 
noids. The action is similar to that of a centrifugal pump. The heavy 



309 



Section 11-08 



masses near the axis are prevented from sinking by the " centrifugal 
suction " arising from the stronger intensity of the vortex aloft. When 
the action of the solenoids and the action of the centrifugal suction have 
the same intensity everywhere in the vortex, no meridional circulation 
will arise, and the motion remains steady. The balance of the vortex is 
thus established by a complete adjustment between the mass field and 
the field of motion. It is important to note that the balance is controlled 
only by the variation in angular speed parallel to the axis. The varia- 
tion of angular speed in the radial direction may be arbitrary. 

Pole 



Warm 




Warm 



FIG. ll-07c. Dynamic balance of zonal motion. 



We may finally consider the case where the atmosphere initially moves 
as a circular vortex, but where the initial distribution of angular speed 
and mass is such that the equation (3) is not satisfied. This vortex is 
unbalanced and a meridional circulation will result in the direction of the 
dominating effect. When the solenoidal effect dominates, that is, when 
the heavy masses are raised too much in the polar region, a " direct " 
circulation around the solenoids results. When the centrifugal suction 
dominates, that is, when the angular speed increases too rapidly with 
" height," a " retrograde " circulation against the solenoids results. 
During the subsequent meridional circulation both the solenoidal effect 
and the centrifugal effect are modified, and the question arises whether 
the motion approaches the steady state of a balanced vortex. This prob- 
lem is intimately connected with that of the stability of the circular 
vortex. It can be shown that the stability depends primarily upon the 
variation of angular speed in the radial direction. 

11 08. Thermal wind in zonal motion. The rules for the dynamical 
balance of (absolute) zonal motion are in qualitative agreement with the 
rules derived in section 8-03 for the variation of the wind with height. 
In the vortex with the cold core, where the isobaric temperature gradient 
is directed towards the pole, the west wind increases with " height/' so 
the shear of the wind is directed toward the east in accordance with the 



Section 11-08 



310 



thermal wind formula. It is possible to derive a thermal wind formula 
for the shear of the wind in zonal motion directly from the circulation 
theorem, 11'07(1). For this purpose the solenoid number is replaced 

by the pressure integral 11*05(1); 

thus 




(1) 



, . 

- / o,*R-Sr=- / aS 

J C JC 



FIG. 11-08. 



The theorem is applied to an infin- 
itesimal meridional curve, consisting 
of two isobaric curve elements (2 
and 4), and two equilibrium line ele- 
ments (1 and 3) parallel to the axis, 
_ as shown in fig. 11-08. The sense 

. . , . . . , of the integration is indicated by the 
Evaluation of thermal wind " J 

in zonal motion. numbers. Both integrals m (1) are 

easily evaluated for this curve. 

The pressure integral vanishes along the isobaric elements 2 and 4, 
and the equilibrium line elements give the contribution 



(2) - 



Here 5s p is the length, measured southward, of the isobaric elements 2 
and 4. Since (2 ) is taken along equilibrium lines we have from 1 1 -03 (5a) 
d<t> a =* -adp, where d<f> a is the variation in the potential of gravitation 
from the lower to the upper isobar along the equilibrium lines 1 or 3. 
The potential of the centrifugal force is constant along the equilibrium 
lines, so from 640(6) 5</ a = & where 5</> is the corresponding variation 
in geopotential along the equilibrium line. Introduced in (2) this gives 



(2') 



f 1 /& 

- / a8p -( 

Jc a\65 



The acceleration integral on the left in (1) vanishes along the equilib- 
rium lines. The isobaric curve elements (6r 2 - -6r 4 = 5r p ) give the 
contribution 



(3) 



- / t 

J c 



The vector element 6r p is directed southward along the isobaric layer. 
Let <f> p denote the angle between the isobar and the axis of the earth. 



(3') - / c^R-5r = 2co a sin <p p R (f~ ) 

JC \0<t>/R 



311 Section 11-09 

Then R*Sr p = Rf>s p sin <p p , so (3) takes the form 

\ 

d(t>8s p . 

Here Rdu a = 8v is the variation in linear velocity along the equilibrium 
line from the lower to the upper isobar. When (2') and (3') are intro- 
duced in (1), we obtain 

(4) 2o> a sin <p p 



a\ds/p 

This formula expresses quantitatively the result which was derived in 
the previous section: If the specific volume increases southward along 
the isobaric layer (vortex with cold core) the west wind increases with 
the distance from the equatorial plane. 

Equation (4) gives the rate of increase of the west wind with the dis- 
tance from the equatorial plane as a function of the rate of inflation of 
the isobaric layer. This formula is similar to equation 8-03(7), which 
gives the shear of the geostrophic wind with increasing height. However, 
whereas 8-03 (7) and the subsequent thermal wind formulas are approxi- 
mate, based upon the assumption that the hydrostatic equation is valid, 
equation (4) is an exact result. Since o> never departs appreciably from 
fi, and the isobars are very nearly horizontal, we have approximately 
o> a sin <p p fi sin <p = tt z . Assuming further that the velocity variation 
can be measured along the local vertical, instead of along the direction 
parallel to the axis of the earth, we have from (4) 

2Q5p !&*_ 1_2>T 

which agrees with the approximate thermal wind equation 8-05(1). 

11*09. Circulation theorems in developed form. The primitive 
circulation theorems are valid for arbitrary closed curves, when the 
integration along the curve is performed at a fixed time. The analysis of 
the mass distribution in zonal motion offers only one example of the 
numerous problems of a similar nature which may be successfully inves- 
tigated with the aid of these theorems. The theorems reveal various 
characteristics of the instantaneous situation in the atmosphere. The 
primitive circulation theorems can therefore be characterized as diag- 
nostic theorems. 

More specialized theorems can be obtained by application of the cir- 
culation theorems to individual fluid curves consisting of the same fluid 
particles at all times. These individual circulation theorems make it 



Section 11-09 312 

possible to estimate changes in the motion from one instant to the next, 
and can therefore be characterized as prognostic theorems. 

1 1 10. Transformation of the acceleration integral for closed individ- 

ual curves. An individual curve is defined as a curve which, once 

chosen, will always later consist of the same fluid particles. Let fir be a 

^ vector line element of such a curve. This 

d* ^rjf, / A M element will move with the fluid and during 

- ~*~~~T I** the motion will generally change its length 

vdtl /(v+ 8 v) dt and orientation. If fir is the vector element 

- -- ,/' * --- . * (see fig. 1140) at the time t, its position at 

the time / -f dt is determined by the displace- 
ments of its endpoints during the time ele- 

ment dt. It follows directly from the diagram, when we follow the two 

vector paths A to B, that 

d 

(1) vdt 4- fir + T (fir)* = fir + (v + to)dt. 

dt 

Hence 

(2) | (ar) = Sv, 

where 5v is the variation of v at the time t from the initial to the terminal 
point of the vector element fir. The result (2) also follows directly from 
the fact that the two operations d/dt and 5 are independent and therefore 
interchangeable. 

With the aid of (2) the acceleration integral in the primitive circula- 
tion theorem 11 '04 (la) can be transformed. The integral element in 
this integral may be developed as follows : 

(3) v-fir = j (v-fir) - v- j (fir). 

dt dt 

Substitution from (2) in the last term on the right gives 
(3') v-fir - I (v-Sr) - vfiv = | (vfir) - 5 (^ 

The last step is evident, since fi(v 2 ) = 5(vv) * 2vfiv. When (3 ; ) is 
integrated around an arbitrary individual curve, the integral of the total 
differential fi(*> 2 /2) vanishes, and we have 



The geometrical summation f c and the time differentiation d/dt are inde- 



313 Section 11-11 

pendent for the same reason as in (2) and are therefore interchangeable. 
The last integral in (4) is by definition the circulation C. Using this 
notation (4) becomes 



This important theorem was derived in 1869 by Lord Kelvin. It can 
be stated as follows: The circulation integral of the acceleration taken 
around a closed individual fluid curve is equal to the rate of change of the 
circulation of the curve. From this theorem Kelvin derived another 
theorem concerning the physical nature of the circulation. 

11-11. Individual circulation in absolute motion. Kelvin's theorem 
holds both for absolute and relative motion. We shall apply it to the 
circulation theorem for absolute motion, 11-05(2). Substituting here 
from 11-10(5), we find for a moving fluid curve 

(!) f-AW 

This is the theorem of individual circulation in absolute motion. 

The theorem was originally developed by Lord Kelvin for the special 
case of a fluid whose density is a function of the pressure only, p = p(p). 
Accordingly the density and, hence, the specific volume are constant on 
every isobaric surface. The isobaric and the isosteric surfaces coincide 
throughout the field, so no solenoids exist. Such a fluid is called auto- 
barotropic (its mass field is automatically determined by the pressure 
field). In general, a fluid like the atmosphere is not autobarotropic, 
since the density depends not only upon the pressure but also upon the 
temperature and the humidity. Since an autobarotropic fluid contains 
no solenoids, theorem (1) reduces to 

(2) ^=0 or Ca=C a0 . 

Cao is the initial circulation of the fluid curve and C a its circulation at an 
arbitrary later instant. Theorem (2) states: The absolute circulation of 
a closed fluid curve is conserved in an autobarotropic fluid. This theorem is 
the hydrodynamical equivalent of the law of conservation of angular 
momentum in elementary mechanics. 

The complete theorem (1) was later developed by V. Bjerknes (1898) 
and can be stated as follows : The rate of change of the absolute circulation 
of a closed individual fluid curve is equal to the number of pressure-volume 
solenoids embraced by the curve. This theorem is equivalent to the 



Section 11-11 314 

theorem in the mechanics of rigid bodies stating that the rate of change 
of the angular momentum of a system is equal to the resultant moment 
of the acting forces. 

11-12. Circulation of the latitude circles in zonal motion. To 

illustrate the circulation theorem, 11-11 (1), we shall examine the changes 
of the zonal circulation in a balanced circular vortex, when the vortex is 
disturbed by a symmetric perturbation. It was shown in section 
11-07 that the solenoids in the circular vortex are annular tubes around 
the axis of the earth. Let the vortex be disturbed by a vortex ring per- 
turbation, such that the particles on each parallel circle are given merid- 
ional impulses of equal strength and direction. This perturbation will 
not destroy the axial symmetry of the motion. Fluid curves which 
coincided with the circles of latitude before the perturbation are still 
circles of latitude after the perturbation, expanded from their original 
position at certain levels and contracted at other levels. Consequently 
the solenoids will remain annular tubes about the axis. So the expand- 
ing and contracting circles will never embrace solenoids and, from the 
theorem 11-11(1), their absolute circulation remains individually con- 
stant. Thus 

/i r* 

(1) - = or C a - const. 

at 

The absolute circulation of the latitude circle in zonal motion is 

(2) C a - / v 6r - 2wRv a - 27rR 2 w a . 

J c 

Substituting this expression in (1) we find 

(3) R 2 ua - const. 

This result resembles the law of conservation of angular momentum for 
a single particle. It could have been derived directly from this princi- 
ple, for the moment about the axis of the forces acting on the fluid circle 
is zero. Equation (3) gives the simple rule that the absolute angular 
speed of an expanding or contracting circle changes in inverse proportion 
to the square of its radius. 

11-13. Relation between absolute and relative circulation. The 
absolute circulation and the relative circulation are defined by the 
expressions 



(1) 



i - / v a '5r, C - / 

J c J c 



315 



Section 11-13 



The absolute and relative velocities are related by 



6-16(3) v 

where r is the position vector from an origin on the axis of the earth. 
We perform scalar multiplication of the three vectors in this equation 
by the vector line element dr of a closed fluid curve c and integrate 
around the curve. Introducing the expressions (1) we find 



(2) 



C + 



xr5r. 



The integral on the right is the absolute circulation of the curve con- 
sidered momentarily fixed to the earth. In the scalar triple product 
under the integral sign the dot and the cross may be interchanged (see 
section 6-14): Q x r*5r = QT x dr. Each integral element is then the 
scalar product of the constant vector Q and the vector r x 6r, so Q can 
be taken outside the integral sign. Thus 



C+Q- rx5r. 



(3) 



This integral is the vector sum of the vector elements r x Sr taken around 
the curve. When the curve c lies in a plane the geometrical meaning of 
this vector integral is obtained as follows. Let the plane of the curve 
shown in fig. ll'ISa intersect the axis of the earth at the point 0, which 




FIG. 1 1 1 3a. Vector area of closed curve. 

may be chosen as the origin of the vector r. The vector product r x dr 
is directed normal to the plane and is numerically equal to the area of the 
parallelogram formed by the two vectors. Thus the vector -J-r x dr has 
the magnitude of the shaded triangular area defined by the two vectors. 
The integral of the vector ^r x dr taken along the upper branch of the 
curve from the point 1 to the point 2 is therefore a vector normal to the 
plane of the curve, having the same sense as the integration, and numeri- 
cally equal to the sectorial area under the branch 1 -> 2 of the curve. 



Section 11-13 316 

The integral from 2 to 1 along the lower branch of the curve is a vector 
with opposite sense, having the magnitude of the sectorial area under 
the branch 2 -> 1 of the curve. The total integral around the whole 
curve is therefore a vector normal to the plane of the curve, with the 
same sense as that of the integration, and numerically equal to the area 
enclosed by the curve. This vector is called the vector area of the curve 
and will be denoted by A, where A is the enclosed area. Accordingly 
we have 



(4) A--J / rx5r. 






This result, here derived for a plane curve, can be shown to hold generally 
for any skew curve, where r may be taken from an arbitrary origin. A is 
then a vector normal to the plane on which the projection of the curve 
encloses a maximum area, and is numerically equal to this area. When 

(4) is introduced in (3), we have 

(5) C a C+20-A. 

This equation gives the relation between absolute and relative circula- 
tions on the earth. 

Equation (5) can be verified directly for the special case of zonal 
motion. Consider one of the latitude circles with radius Jf?, and per- 
form the integration in the circulation integrals from west to east. The 
absolute circulation is 

(6) C a - / v-6r - 2wRv a - 2irR 2 a . 

Jc 

And the relative circulation is 

(7) C = / v-5r - 2wRv - 2irJ8 2 . 

J c 

The vector area A of the latitude circle is parallel to the axis of the earth, 
directed toward the north, and its magnitude is irR 2 . Thus 

(8) 2Q'A - 2irR 2 tt. 

Substituting the three expressions (6, 7, 8) in (5) and dividing out the 
factor 27rR 2 , we find co a = co + fi, which verifies the theorem. 

The sign of the scalar product in (5) is determined by the angle 
between the two vectors and A. The sense of the vector A is deter- 
mined by the sense of integration along the curve, which is open for 
choice. We shall now choose the sense of integration so that it has the 



317 



Section 11-14 



same sense as the rotation of the earth, as shown in fig. 11-136. With 
this choice the angle between Q and A is always acute (except when the 
plane of the curve is parallel to the axis of the earth). Let S be the 
positive area enclosed by the equatorial projec- 
tion of the curve. The scalar product Q*A is 
then always a positive quantity equal to OS, 
which introduced in (5) gives 



(9) 



C+ 




FIG. 11-136. Equatorial 

projection of area of closed 

curve. 



Since the sense of integration has been chosen, a 
unique sign convention for the circulation is 
introduced: Circulation with the same sense as 
the rotation of the earth is positive, and circu- 
lation with the opposite sense is negative. For 
horizontal curves which do not intersect the 
equator the circulation is then cyclonic in 
both hemispheres if it has the same sense as 
the rotation of the earth, and anticyclonic if it has the opposite 
sense. 

11 -14. Individual circulation relative to the earth. The theorem for 
the change in relative circulation is obtained by time differentiation of 
equation 11-13(9); thus 

dC a dC d2 

~dT = lu + 2n lu' 



Substituting here for dC a /dt from 11-11 ( 1 ) and solving for dC/dt, we find 

dC d% 

- # a ,_ p -212 . 



(1) 



With the chosen sense of integration the solenoidal term is positive when 
the sense (V V/0 of the solenoids is the same as the rotation of the 
earth, and is negative when the solenoids have the opposite sense. 

Equation (1) is the theorem of circulation relative to the earth of an 
individual fluid curve. It is the most important of the circulation theo- 
rems for the study of atmospheric motion and is in usual meteorological 
language referred to simply as the circulation theorem. It can be stated 
as follows: The rate of change of the circulation relative to the earth of an 
arbitrary closed fluid curve is determined by two effects: (i) the solenoid 
effect will tend to change the circulation in the sense of the solenoids by an 
amount per unit time equal to the number of solenoids embraced by the 
curve, (ii) The inertial effect will tend to decrease the circulation by an 



Section 11-14 318 

amount per unit time proportional to the rate at which the projected area of 
the curve in the equatorial plane expands. 

This theorem holds for any closed fluid curve if the effect of friction is 
neglected. By appropriate choice of the curve many types of atmos- 
pheric motion which are too complicated for complete analytical treat- 
ment can be examined qualitatively. 

It should be noted that the inertial term vanishes when the relative 
motion is zero. So the primary origin of the circulation (both absolute 
and relative) is the dynamic action of the solenoids. The effect of the 
inertial term is to modify relative circulation which already exists. This 
modification is of importance for every large-scale motion of the atmos- 
phere. 

The significance of the inertial term becomes clear when we consider 
curves which embrace no solenoids. This will be the case for any curve 
in an autobarotropic fluid, but the results will hold with rough approxi- 
mation for horizontal curves, since in general few solenoids intersect the 
horizontal levels. In this case equation (1) may be integrated, and the 
circulation theorem becomes 

(2) C- C = 2fl(Z -S). 

This equation gives the following rule : A closed fluid curve, moving from 
one position into another, gains (or loses) an amount of circulation which 
is proportional to the decrease (or increase) in the area enclosed by the 
equatorial projection of the curve. For horizontal curves which do not 
intersect the equator the rule is simply: The curve gains cyclonic circula- 
tion while its equatorial projection contracts, and it gains anticyclonic 
circulation while its equatorial projection expands. 

11 -IS. Circulation of the circles in a local vortex. The rule at the 
end of the preceding section explains in a qualitative way how cyclonic 
and anticyclonic circulations are generated in the atmosphere. It is a 
well-known empirical fiact that a cyclone with closed isobars in the 
lower layers has horizontal convergence toward the central region, and 
consequently ascending motion with clouds and precipitation in the 
central part of the cyclone. Therefore all closed horizontal curves 
embracing the cyclone center contract and gain cyclonic circulation. 
Similarly the air over an anticyclone subsides and spreads out hori- 
zontally in the lower layers. So horizontal curves embracing the anti- 
cyclone center will expand and gain anticyclonic circulation. 

Although these qualitative rules for the generation of cyclonic and 
anticyclonic circulations are similar, there is an important difference 
between the two processes when they are compared quantatively. To 



319 



Section IMS 



demonstrate this difference we shall examine the following idealized 
model: The cyclone (or anticyclone) is assumed to have circular con- 
centric streamlines with a uniform speed on each streamline. This 
model may be called a local circular vortex. We shall further assume that 
symmetrical convergence toward (or divergence from) the central region 
occurs in the vortex, and examine the change of circulation of the con- 
tracting (or expanding) fluid circles. If no solenoids intersect the hori- 
zontal levels, the circulation of the fluid circles changes in accordance 
with the theorem 11-14(2); thus 



(1) 



C-f 2QS - const. 



Although the result will be valid in both hemispheres, we shall in the 
following restrict the discussion to a local vortex in the northern hemis- 
phere. Let R denote the radius of an arbitrary circle in the vortex, and 
let co be the relative angular speed on this circle. The circulation around 
the circle is then 



O\ / 9-.J?2 /% 

(2) C = ZTTK co. 

Positive values of co mean cyclonic circulation 
and negative values mean anticyclonic circula- 
tion. The area enclosed by the circle is A = 
TrR 2 , and the equatorial projection of the area 
(see fig. IMS) is 



Axis of 
local vortex 

V 



(3) 



S - A sin <p = 



sin <p, 



where <p is the latitude of the axis of the vor- 
tex. Substituting these values of C and S in (1) 
and dividing out the constant factor 2?r, we find 




(4) 



R 2 (w+ 



const. 



Equator 

FIG. IMS. Equatorial 

projection of circle in local 

vortex. 



In the special case where the axis of the local vortex coincides with the 
axis of the earth this equation is identical to 11*12(3). Equation (4) 
gives the corresponding law for a vortex in arbitrary latitude: On a 
given fluid circular streamline, which expands or contracts because of 
symmetric horizontal divergence or convergence, the quantity (co + $2,) 
is inversely proportional to the square of the radius. 
f- If the vortex has horizontal convergence toward the axis, the circles 
contract and co increases, so the vortex gains cyclonic circulation. As 
the convergence continues, the radius of the contracting circle ap- 
proaches zero, and from (4) co + fi z - > , or co -* < . Thus the cyclonic 
angular speed will increase indefinitely if the convergence continues to 
operate. 



Section 11-15 



320 



If the vortex has horizontal divergence out from the central region, 
the circles expand and w decreases, so the vortex gains anticyclonic circu- 
lation. If the expansion were to continue until the radius approaches 
infinity, we should have from (4) co + Sl z -> 0, or w -> - 12^. Actually the 
expansion can never proceed so far, so the anticyclonic angular speed 
which is generated by horizontal divergence is always numerically smaller 
than the critical value 12 Z . This limiting value corresponds to the 
anticyclone with the maximum strength of the horizontal pressure field, 
as was shown in section 7-2 1. 

11*16. Vorticity. Any horizontal area A bounded by a closed curve 
may be divided by two families of curves into infinitesimal elements dA, 
as shown in fig. 11-16. The sum of the circulations around the bound- 
aries of these elements, taken all in the 
same sense, is equal to the circulation 
around the original boundary of the whole 
area. For in the sum the procession along 
each side common to two elements comes in 
twice once for each element, but with 
opposite sense and therefore disappears 
from the result. There remain then only 
the processions along those sides which are 
part of the original boundary. Thus, if C 
denotes the circulation around the original boundary of A, and dC 
denotes the circulation around an arbitrary element 8A , we have 




FIG. 11-16. 



Addition of circu- 
lation. 



C= / 3C, 



(i) 



where the summation is extended over all the elements $A. 

The limit of the ratio of the circulation BC around an infinitesimal ele- 
ment to the area 8A of that element is called the vorticity, and is denoted 
by f . Thus 

'-a- 

In a rough sense the vorticity is the circulation around unit horizontal 
area. When 8C is eliminated from (1) by means of (2), we have 



(3) 



L 



This theorem, given by Stokes (1854), states: The circulation around a 
closed horizontal curve bounding any finite area A is equal to the integral 
of the vorticity taken over the area A. 



321 Section 11-18 

The two-dimensional theorem stated here is only a special case of 
Stokes's theorem. The general three-dimensional theorem has the 
same form as (3) and is valid for an arbitrary surface in space bounded 
by a closed curve. The vorticity is then a vector and the scalar quan- 
tity f, as defined by (2), is the component of the vector vorticity along 
the normal of the surface element 8A . For horizontal areas and curves 
the quantity f, which in the following will be called the vorticity, is 
actually the vertical component of the vector 
vorticity. 

11-17. The vorticity in rectangular coordi- ' s A 

nates. Let the horizontal element of area y 

be the infinitesimal rectangle 8A = 8x8y shown x x+dx 

in fig. 1M7. The circulation 8C around this FIG n ^ 

element, taken in the sense of the positive z 

axis (that is, with positive cyclic sense), is the sum of the processions 
along each of the four sides: 

/ &Uj, \ ( &u x \ 

8C ~ v x 8x + ( v y + 8x ] 8y - [ v x + 8y ] 8x - v y 8y, 
\ d* / \ by J 

which reduces to 

(1) 8C> 

Dividing both sides by the area 8A = 8x8y, we have from 1116(2) that 

8C fay dv x 



Equation (2) gives the analytical expression for the 
vorticity in rectangular coordinates. 

1 1 1 8. The vorticity in natural coordinates. Next 
let the horizontal surface element be the infinitesimal 
area defined by two streamlines and two straight 
lines normal to the lower streamline, as in fig. 11 18. 
In the figure the two normals are extended in the 
horizontal plane to their point of intersection, which 
is the center of horizontal curvature of the lower 
streamline. Let v be the speed on the lower stream- 
line, and let s, n be natural coordinates, as in 7-02. 

The circulation around this surface element, taken in the positive sense 

as shown by the arrows, is 




8C - vR s ty - [ v + 8n ) (R s - 



Section 11- 18 322 

When the multiplication is performed, this expression reduces to 

(1) SC = (-J- - ^ + ~ ^ 8n] R s Wn. 

\Rs dn R s dn / 

The third term in the parentheses approaches zero with 8n. Thus, when 
both sides of the equation are divided by the area of the element, 
bA - Rstybn, we have 

(2) f-rz- 1 ^-;?' 

dA On 

which is the analytical expression for the vorticity in natural coordi- 
nates. According to this formula the vorticity manifests itself at any 
point in a horizontal current by the curvature of the streamlines, or by 
the horizontal rate of shear, or both. In absence of shear the vorticity 
has the same sign as the curvature. In a straight current the vorticity 
is positive if the speed increases to the right of the current, and negative 
if the speed increases to the left. 

In the special case where the motion is a constant rotation with the 
angular speed w about a vertical axis, the streamlines are concentric 
circles. The speed is here v = wUs, and the rate of shear is cto/dw = 
du/dlZs = o>. Substituting these values in (2) we find 

(3) f-2. 

In the field of constant rotation the vorticity is constant throughout and 
is equal to twice the angular speed of rotation. It can be shown that the 
vector vorticity in this case is a vector along the axis and is equal to 
twice the angular velocity. 

11-19. Absolute and relative vorticity. The relation between abso- 
lute and relative circulations is given by the equation 11*13(5). If the 
curve is the edge of an infinitesimal horizontal surface element dA, and 
the absolute and relative circulations are denoted respectively by dC a 
and BC, we have 

(1) C-C+21HA. 

Here 5A is the vector area of the surface element. Defining the circu- 
lation as positive when it has the same sense as the positive vertical, 
we have in both hemispheres 5A &4k. Substituting this in (1), and 
dividing the whole equation by 8A, we find that 

(2) ra-r+2o-k 



With the sign convention chosen here, the absolute and relative vorticities 
are positive in both hemispheres when their sense is that of the positive 



323 Section 11-20 

vertical. This sign convention is in accordance with the one intro- 
duced in chapter 7 for the angular speed. Accordingly, cyclonic vortic- 
ity is positive in the northern hemisphere and negative in the southern 
hemisphere. It should be noted that this sign convention is different 
from the one used for the circulation in 1 1 -13 (9) and subsequently for the 
circulation theorem in 11-14(1). In those formulas the circulation with 
the cyclic sense of the earth's rotation was positive. That gave the 
simplest rules for circulation of curves with an arbitrary orientation 
in space. In the northern hemisphere the two conventions are the same ; 
in the southern hemisphere they are opposite. 

If the atmosphere is at rest, the relative vorticity is everywhere zero. 
The absolute vorticity about the vertical is in this case 2S2 Z , and the 
vector vorticity can be shown to be 212; see 11-18(3). In the general 
case the absolute vorticity is the sum of the vorticity relative to the 
earth and the absolute vorticity of the earth. 

1120. The theorem of absolute vorticity. From the theorems of 
circulation of individual moving curves equivalent theorems for the 
vorticity of individual moving particles may be derived. Consider an 
infinitesimal horizontal surface element 5A in a horizontal or nearly 
horizontal current. If the surface element is part of the fluid, its 
boundary will remain a closed curve moving with the fluid, and its 
absolute circulation 5C a will change in accordance with the general 
theorem 11-11(1). Since the element is horizontal , the solenoids may be 
neglected, and the theorem takes the form 

(i) I (c.) = o. 

Since SA is infinitesimal, we have from 11-16(2) 

(2) SCa-T^M-M*. 

8A 

Substitution of (2)*in (1) gives 

J f 

at 
or, when the differentiation is performed, 

(3) 
v 



f a dt 5A dt 
The second term is the rate of horizontal expansion of the moving ele- 



Section 11-20 324 

ment per unit area. From 10-03(3) this is the horizontal divergence 
V#*v. Hence equation (3) can be written 

This theorem is equivalent to the theorem 11-11 (2) for horizontal motion. 
It states that the rate of change of the absolute vorticity of a moving 
element is proportional to its horizontal divergence. It should be noted 
that the theorem is exact only when the motion is strictly horizontal and 
the fluid autobarotropic. However, all large-scale atmospheric currents 
are so nearly horizontal, and the number of solenoids which intersect the 
horizontal levels are so few that the theorem can be used with sufficient 
accuracy at levels above the layer of friction. The theorem (4) is a 
special case of the famous vorticity theorem of Helmholtz (1858). The 
physical significance of Helmholtz' theorem, which involves the vector 
vorticity, is in every respect equivalent to that of the circulation theorem, 
11-11(1). 

11-21. The theorem of relative vorticity. When the expression 
11-19(2) for the absolute vorticity is substituted in the theorem 11-20(4) 
we have 

d 

(1) - (f + 2Q Z ) - - (f + 2a z )V//'V. 
dt 

This is the theorem of relative vorticity. For horizontal motion it is 
equivalent to the theorem of relative circulation 11-14(2) and can also be 
derived directly from this theorem. The theorem (1) was derived by 
Rossby (1938) and has been used extensively by him and others for the 
analysis of the flow pattern in stationary and moving atmospheric 
waves. 

Although the circulation theorem has a much simpler form and can be 
stated in brief and precise terms, the vorticity theorem has great advan- 
tages, particularly for the investigation of the structure of horizontal flow. 
The main reason is that the vorticity can be expressed analytically in 
terms of the velocity components, as in 11-17(2), or in terms of the 
curvature and shear of the flow pattern, as in 11-18(2). 

The vorticity theorem (1) takes a convenient form when the differen- 
tiation is performed on the latitude term 2$l z . We have 

(2) j (2Q 2 ) - 2Q (sin ^>) =* 2Q cos <p 

Here d<? is the change of latitude of the moving particle during the time 



325 Section 11-22 

element dt. Introducing the linear meridional displacement dy = ad<p, 
we have 

f3 ) *? . A ?y = ^ . 

^ ' dt a dt a' 

This is valid in both hemispheres when the y axis points to the north (the 
standard system), and when <p is counted positive in the northern hemi- 
sphere and negative in the southern hemisphere. Substituting (3) in 
(2), we have 

d , 212 cosv? 

(4) - (20.) = - 5? v 
dt a 

The vorticity theorem (1) can therefore be written 

df 217 cos <p 

(5) = - - ~v y - (f + 2Q,)Vjyv. 
at a 

If the horizontal flow is non-diverging, equation (5) becomes 

d 2 12 cos <p 



The factor (2Q cos ^>)/a is positive in both hemispheres, so the vorticity 
decreases when the particle moves north and increases when the particle 
moves south. This rule is equivalent to the result already obtained in 
section 11 44 from the circulation theorem: When the particle, con- 
sidered as a small horizontal disk, moves toward the pole its equatorial 
projection increases, and it gains anticyclonic vorticity. When the 
particle moves toward the equator its equatorial projection decreases, 
and it gains cyclonic vorticity. 

11-22. Air current crossing the equator. As an example of the use 
of the vorticity theorem, consider a non-diverging current crossing the 
equator. Near the equator we have approximately cos^>=l, so 
11-21(6) becomes 

m *--*?, 

(1) dt' a v 

If the current crosses the equator from south to north, the vorticity of 
the particles moving with the current will decrease in both hemispheres. 
If the vorticity is zero at the southern origin of the current, the particles 
will arrive in the northern hemisphere with negative vorticity, which 
here is observed as anticyclonic. If the current crosses the equator from 
north to south, the particles will arrive in southern latitudes with posi- 



Section 11-22 



326 



tive vorticity, which here means anticyclonic vorticity. Therefore air 
which has newly crossed the equator has a tendency to show anticy- 
clonic vorticity. According to 11'18(2) this vorticity shows up in the 
current as anticyclonic curvature or anticyclonic shear. In a broad 
current the shear is generally small, and the vorticity appears mainly as 
anticyclonic curvature of the current. A good example of this effect 
is found in the summer monsoon in India. As the thermal low develops 
over the Asiatic continent, a branch of the south-east trade wind of the 
southern hemisphere is forced to bend northward. As this current 
crosses the equator, it bends anticyclonically and arrives over India as a 
monsoon from the southwest. 

1 1 23. Air current crossing a mountain range. If a broad and fairly 
straight current of air crosses a mountain range, it is well known from 
weather maps that a trough develops in the streamlines on the lee side of 
the mountain. This deformation of the current is easily explained in a 
qualitative sense from the vorticity theorem. 




FIG. 11*23. Air current crossing a mountain range. 

For simplicity it will be assumed that the current is a straight zonal 
current flowing from west to east, and that there is no horizontal shear 
in the current. Let this current be obstructed by a mountain range 
running normal to the current, as in fig. 11-23. Although the results 
will be valid in both hemispheres, we shall consider only the conditions 
in the northern hemisphere. We shall neglect the influence of friction 
in the surface layer and shall study the pure inertial effect caused by the 
presence of this obstacle. An air particle near the ground will flow 
uphill on the windward side and descend on the lee side of the mountain. 



327 Section 11-23 

If the stratification of the air is stable, the flow at higher levels will be 
less modified by the obstacle, and above a certain level the deformation 
is negligible. The air below this level is subjected to vertical shrinking 
and horizontal divergence during its approach to the crest of the moun- 
tain, and to vertical stretching and horizontal convergence during its 
descent on the lee side of the mountain. The corresponding changes of 
the vorticity during the crossing are given by the vorticity theorem 
11-21(5): 



dt a y 

Since the current is zonal, v y 0, and the theorem reduces to 

d 
(2) (r ~H 2 2 sin ^)V//*v 

Hence the air particles gain anticyclonic vorticity on the windward side 
of the mountain, where the horizontal divergence is positive, and they 
gain cyclonic vorticity on the lee side of the mountain, where the hori- 
zontal divergence is negative. Since the current arrives at the obstacle 
without curvature or shear, and hence without any vorticity, the air 
particles acquire increasing anticyclonic vorticity during their climb. 
They pass over the crest of the mountain with a maximum of anti- 
cyclonic vorticity, which they subsequently lose during their descent on 
the lee side. 

If the current and the mountain have infinite lateral extent to the 
north and to the south, no horizontal shear develops during the crossing. 
Thus the vorticity can show up only as a change of curvature of the 
current as indicated in fig. 11 -23: increasing anticyclonic curvature on 
the windward side, maximum of anticyclonic curvature on the crest, 
and decreasing anticyclonic curvature on the lee side. 

If equation (2) were valid during the entire crossing, the particles 
would lose exactly the same amount of vorticity during the descent as 
they gained during the ascent, and the current would leave the mountain 
as the straight current indicated by the broken streamlines. However, 
as soon as the anticyclonic turning begins, the particles obtain a velocity 
component toward the south, and the subsequent deformation of the 
current is controlled by the complete vorticity theorem, equation (1). 
Since v y < 0, the latitude term is positive during the entire crossing. It 
counteracts the effect of the divergence until the crest is reached, but it 
cannot reverse the sign of this effect. Beyond the crest the latitude 
term and the divergence term have the same sign. The particles thus 
gain more vorticity during their descent than they lost during the ascent, 



Section 11-23 328 

and the current will leave the mountain with cyclonic curvature, as 
indicated by the full streamlines in the figure. 

In the absence of other kinds of divergence beyond the mountain 
range, the vorticity is from then on controlled only by the change in 
latitude. As a consequence the current gains cyclonic vorticity and 
curvature as long as the particles move south, that is, until the wind 
becomes a pure west wind. At this point the current has a maximum of 
cyclonic curvature and will consequently bend north. As the particles 
move north they lose their cyclonic vorticity and subsequently gain 
anticyclonic vorticity, so that the current eventually bends back anti- 
cyclonically again, and so on. 

The effect of the mountain range upon an air current crossing it is thus 
to generate a stationary wave in the current beyond the mountains, 
beginning with a cyclonic trough immediately behind the mountain. 
Only this first cyclonic trough is usually well developed on the weather 
maps. The rest of the wave train is probably damped out by friction 
and other kinds of horizontal divergence which have not been considered 
here. 

The stationary wave which develops in a current crossing a mountain 
range must evidently satisfy the tendency equation 10-07(3). Since the 
wave is stationary the pressure tendency must be zero throughout; thus 



(3) 



00 

/ 



The vertical motion at any point must have just the right value to 
balance the effect of divergence in the vertical column above the point. 
In a qualitative sense it is seen that this condition is satisfied. The 
upslope wind on the west side of the mountain will cancel the effect of 
the horizontal divergence in the vertical column above, and likewise the 
downslope wind on the lee side of the mountain will cancel the effect of 
the horizontal convergence in the vertical columns in this region. 

11*24. Non-diverging wave-shaped flow pattern. In section 10-10 
we made use of the concept of transport capacity in isobaric channels to 
examine the conditions for longitudinal mass divergence in wave- 
shaped patterns. In particular, we obtained the critical condition for 
zero longitudinal mass divergence by this transport method. A similar 
result may be derived from the vorticity theorem, as shown by Rossby. 
In this section we shall consider some of the physical implications of this 
vorticity method, and also attempt to bring out the points of similarity and 
difference between the two methods. 



329 



Section 11-24 



Consider a westerly current without horizontal shear, upon which is 
superimposed a wave disturbance of infinite lateral extent. The stream- 
line amplitude is then independent of latitude. In such a wave the 
transversal divergence (section 10-12) is zero, and the total horizontal 
divergence is equal to the longitudinal divergence. We shall assume that 
there exists a level where the velocity divergence is zero and examine the 
conditions at that level. 

We first note that, since both the transversal divergence and the 
longitudinal divergence are zero at that level, the zonal velocity com- 
ponent must be constant throughout the level, having the value v of the 
actual speed at the trough line and the wedge line. Fig. 11-24 shows an 




FIG. 11-24. 

arbitrary streamline at the level of non-divergence, and also the path of 
the particle which at the time of the diagram is located at the inflection 
point of the streamline. The relation between the wave length LS of the 
streamline and the wave length L of the path is obtained as follows: 
Let T be the period during which the particle traverses one complete 
wave length of its path; thus 

(1) L=vT. 

The particle arrives at the northern bend of the path after the time 
^r. The arrival of the particle at this point is simultaneous with the 
arrival of the streamline crest, because the wind must be from the west 
on this meridian at that time. Therefore the wave, moving with the 
speed of propagation c, travels the distance \(L- LS) during the time 
J7*; hence 

(2) L - L s - cT. 

Taking the ratio between (2) and (1) and rearranging, we find 

(3) ^l.Llf. 



Section 11-24 330 

The relation between the amplitude A s of the streamline and the 
amplitude A of the path is obtained by considering the relative stream- 
lines. As stated at the end of section 10*15, the relative streamlines 
are also the relative paths, and they therefore indicate the true meridional 
displacement of the air particles. So the amplitude A R of the relative 
streamline is equal to the amplitude A of the path. From 10-15 (2) 
we have then 



(4) 



Accordingly, the amplitudes of the streamline and the path have the 
same ratio as their wave lengths, the ratio being that of the relative 
zonal wind to the actual zonal wind. 
At the level of non-divergence the vorticity theorem 11-21 (1) becomes 

-(f+2nsin^) = 0. 

According to this theorem the absolute vorticity, f -f 2Q sin^>, remains 
constant for any given individual particle while it moves along the path. 
Since the current has no shear at the trough line and at the wedge line, 
the relative vorticity f at these lines is, from 11-18(2), 



Equating the absolute vorticity of the particle at the southern bend P 
of the path (where the particle passes the longitude of the trough line) 
to its value at the northern bend P f of the path (where the particle passes 
the longitude of the wedge line), we have 

(5) vK s + 20 sin <? P vK' s + 2fi sin <pp. 

This formula resembles 10-10(2') for the transport capacity in a wave- 
shaped isobaric channel. When the longitudinal divergence is zero that 
formula becomes 

(6) Kv + 2Q sin <t> s - K f v + 212 sin w 

<?s and <ps' are the latitudes respectively of the northern and the southern 
bends of the isobaric channel (assumed coincident with the streamline). 
Although the two formulas (5) and (6) are similar, they are the state- 
ments of different physical principles. The vorticity formula (5) is 
satisfied when the horizontal velocity divergence is zero and the current 
is barotropic. This formula contains the curvatures of the streamline 
and equates the absolute vorticity at the southern and northern bends 
of the path at the times when the moving particle occupies these points. 



331 Section 11-25 

The transport formula (6) is satisfied when the longitudinal mass diver- 
gence is zero. This formula contains the curvatures of the path and 
equates the transport capacities at the southern and northern bends of a 
streamline at one fixed time. 

In general the two principles are not equivalent, for zero mass diver- 
gence usually does not mean zero velocity divergence. However, in the 
special case considered here the two formulas (5, 6) are equivalent. To 
show this we first rearrange the terms and write the two formulas as 
follows: 

(7) (Ks KS)V = 212 (sin <p p > sin <p p ) = 412 cos sin o>, 

(8) (K K f )v = 212 (sin <?$' s ^ n <f>s) = 412 cos <p sin 03. 

V is the central latitude of the streamline and the path, and o>, a s are 
the angular amplitudes respectively of the path and the streamline. 
Taking the ratio between (7) and (8), we find 

K - K' sin ff S 

W ~ ^7 = 

/vs &s sin <f P 

The formulas 10'10(6) give the kinematic relations which exist between 
the curvatures of the streamline and the path at the two bends. Substi- 
tuting in (9) for K and K 1 from 10-10(6), we find: 



(10) 



sin <r p 



A comparison of this formula with (4) would seem to indicate that (10) 
is slightly incorrect. The error is however only apparent. As in for- 
mula 10-11 (7), the apparent error comes from a combination of spherical 
and plane methods. The two formulas (5, 6) are valid on a spherical 
level. When they are applied to streamlines and paths in a " plane 
level/' for which the formula (4) is derived, a slight inconsistency will 
result. 

The formula (9) is then correct at a level of non-divergence, and 
accordingly the two formulas (5, 6) are consistent. Later, in chapter 12, 
it will be shown how the condition for non-divergence is derived directly 
from the vorticity theorem. 

11-25. Export. The vector area A of a closed curve was defined in 
section 1 1-13 as a vector normal to the plane of the curve arid numerically 
equal to the area enclosed by the curve. Consider any closed fluid curve 
which moves along in a current. During a time element dt a vector line 
element Sr of the curve sweeps over the vector area vdt x 5r. The rate at 



Section 11-25 332 

which the vector area of the curve changes is accordingly 



This result can also be obtained by differentiation of equation 1 1 -13 (4) . 

We shall apply (1) to any horizontal curve in a horizontal atmospheric 
current. Taking the integration in the cyclic sense of the positive verti- 
cal, we have for such curves that A = Ah, where A is the area enclosed by 
the curve. Performing scalar multiplication in (1) with the vertical 
unit vector k, and interchanging the dot and the cross in the scalar triple 
product under the integral, we find 

(2) ^ 

dt 

Let n be a horizontal unit vector normal to the curve, directed out from 
the region enclosed by the curve, and let 6s be the length of the vector 
element 5r. Then dr xk = nfo, and accordingly 

(3) k x v5r = vSr xk = vnSs = v n ?>s. 

It follows then that only the velocity component v n normal to the curve 
is subjected to the integration in (2). The integral in (2) is called the 
export, and will be denoted by E. Thus by definition 



/dA 
kxv8r- 
dt 



I, 



There is a close mathematical analogy between the export and the circu- 
lation. Both are represented by line integrals around a closed curve, one 
integrating the normal component of the velocity and the other the 
tangential component. 

The export has an additive property similar to that of the circulation, 
1 1 1 6 ( 1 ) . Let the area A which is enclosed by the curve be divided into 
infinitesimal elements 8A by two families of curves, as in fig. 11-16. 
The sum of the exports 8E from these elements is equal to the export E 
through the original boundary of the whole area. For in the sum the 
transport through each side common to two elements comes in twice 
once for each element and therefore disappears from the result. 
There remain then only the transports through those sides which are 
parts of the original boundary, and consequently 



(5) E - / dE. 

JA 

The export dE from the element dA is, from (4), given by 



333 Section 11-26 

d(8A)/dt. And the export per unit area has the limiting value 



The expression on the right is, from 10-03(3), the horizontal divergence 
of the velocity field. When dE is solved from (6) and substituted in 
(5), we obtain: 



(8) E= I V//-V&4. 

JA 

This theorem was derived by Gauss (1813) and states: The export 
through a closed horizontal curve bounding an area is equal to the integral 
of the horizontal divergence over that area. 

The two-dimensional theorem stated here is only a special case of 
Gauss's theorem. The general three-dimensional theorem states that 
the export through a closed surface bounding a region is equal to the 
volume integral of the divergence over that region. 

1 1 '26. Irrotational vectors. A vector a is called irrotational in a given 
region if, for all closed curves in that region, the circulation integral of 
the vector is zero: 



(1) / a-5r=0. 

JC 

We have shown earlier that the circulation integral of a potential 
vector Ve is zero for any closed curve in the field of e. The potential 
vector is accordingly irrotational. We shall now prove the converse 
statement, namely, that any irrotational vector a is potential. That is, 
we shall prove that the vector a can be represented as the ascendent of a 
scalar function s, which is called the potential of a. 

Consider an arbitrary closed curve in the field of the irrotational 
vector a. Any two points P and P on this curve divide the curve into 
two branches. Since the circulation integral (1) is zero, the procession 
integral of a from P to P along one branch is the negative of the pro- 
cession integral from P to PQ along the other branch. The integral from 
PQ to P has accordingly the same value along both branches of the closed 
curve. Since the closed curve is arbitrary, we conclude that the proces- 
sion integral of an irrotational vector is independent of the path. Let 
P be a fixed point and P a variable point. For any path from PQ to P, 
the integral 

(2) e 



Section 11-26 334 

is then independent of the path. The integral e is therefore a function 
only of the position of the variable point P. The total differential of e 
is from (2) 

(3) te - a^r, 

where 5e is the variation of e through the displacement 8r. But, accord- 
ing to 443(1), the total differential of the scalar function e can also be 
written 

(4) 8s = Ve'5r. 

Since (3) and (4) hold for every direction of 6r, we have 

(5) a = Ve, 

which proves the statement made at the outset. 

The irrotational vector has thus two fundamental properties: (i) Its 
circulation integral around every closed curve is zero, (ii) It can be 
expressed as the ascendent (or gradient) of a scalar function, called the 
potential of the irrotational vector. Either one of these properties might 
be used as the definition of irrotational. 

1127. Velocity potential. It was shown in section 11*11 that the 
absolute circulation of individual fluid curves is conserved in an auto- 
barotropic fluid. If the motion of such a fluid is started from absolute 
rest, the circulation of any fluid curve is initially zero and will then 
remain zero during the subsequent motion. The velocity is therefore 
an irrotational vector, and so the motion is called irrotational. For 
such a motion there exists, as shown in the preceding section, a scalar 
function <p> such that 

(1) v a -V^. 

The function <p is called the velocity potential. Therefore irrotational 
flow also is called potential flow. 

The above considerations lead to the following important conclusion : 
If an autobarotropic fluid (without friction) is started from rest, its 
motion is potential at any later time. 

The existence of a potential makes the study of the motion rather 
simple. This is because the potential is a single scalar function, whereas 
the components of the velocity are a set of three scalar functions. The 
study of the motion of autobarotropic fluids has for this reason been 
brought to a high level of perfection in the field of classical hydrodynam- 
ics. However, the atmosphere is in general baroclinic, and the solutions 
of potential flow have therefore only limited interest for meteorology. 



335 Section 11-28 

11-28. Stream function. Another application of the irrotational 
vector, which is useful for the study of atmospheric motion, will now be 
discussed. Consider any horizontal current in the atmosphere, and 
assume that in this current a level exists where the horizontal divergence 
of the velocity is zero. From Gauss's theorem 11 -25 (8) the export 
through any closed curve in that level is zero. Therefore, from 1 1 -25 (4) , 



(1) / kxv6r=0 

Jc 

for every closed horizontal curve in the level of non-diverging flow. The 
horizontal vector k x v is then irrotational at the level of non-divergence 
and, as shown in section 11-26, there exists a scalar function \l/ at that 
level such that 

(2) kxv=-V^. 

The velocity is perpendicular to the vector - V& and is therefore every- 
where parallel to the lines of constant \l/. These lines are consequently 
the streamlines, and the function \l/ is for this reason called the stream 
function. We have then the following important rule: If a level of non- 
divergence exists in a horizontal current, the motion at that level can be 
described by a scalar stream function which is constant along the stream- 
lines. The stream function was introduced in this way by Lagrange 
(1781 ) and is sometimes called Lagrange's stream function. The stream 
function has, from (2), the dimensions 



(3) M - 

The variation of the stream function from streamline to streamline is 
obtained by integration of (2). If P is a fixed point on the streamline 
^o and P is a variable point, the value of ^ at P is given by 

p P 

[kxvdr=[vt 
"~ "J XV ~J Vn S ' 

Po Po 

In the second integral v n represents the velocity component normal to the 
path of integration; see 11-25(3). The variation of the stream function 
fronTstreamline to streamline accordingly measures the velocity trans- 
port in the channel between the two streamlines. That this transport is 
constant all along the channel is just another statement of the fact that 
the flow is non-diverging. 

The existence of a stream function in a surface of non-diverging flow 
holds quite generally without any restriction as to the shape of the sur- 
face. The stream function may always be expressed analytically in 



Section 11-28 . 336 

terms of curvilinear coordinates in the surface. If the levels are con- 
sidered as spherical, the stream function in a level of non-diverging flow 
is most conveniently expressed in spherical coordinates. If only a 
limited region of the earth is considered, the level may in the first approx- 
imation be assumed a plane surface. The stream function can then be 
expressed as a function of the rectangular coordinates #, y of the standard 
Cartesian system. In this case the rectangular components of (2) 
become 



As a check we may substitute these values for the velocity components 
in the Cartesian expression for the horizontal divergence, 

10-03(1) v// . v = ^ + ^. 

bx by 

It is seen that the horizontal divergence is zero when the velocity field 
satisfies the conditions (4). 

The Cartesian expression for the vorticity is given by 

iM7(2) r-!r-!r- 

Ox Oy 

At a level of non-divergence where the velocity field is represented by a 
stream function ^, the vorticity may be expressed in terms of the stream 
function. When the values (4) of the velocity components are sub- 
stituted, we find that 



If the vorticity is equal to zero, (5) becomes Laplace's d ifferentiaf equa- 
tion for two dimensions. Accordingly, the flow at the level of non- 
divergence is irrotational if the stream function satisfies Laplace's 
equation. 



CHAPTER TWELVE 
THEORY OF WAVES IN A ZONAL CURRENT 

12*01. The atmospheric equations. The final goal of dynamic 
meteorology is the theoretical prediction of the weather, that is, from 
dynamic theory to determine the state of the atmosphere at some future 
time when its initial state is known. We are still far from the solution 
of this most general problem in atmospheric dynamics. 

According to a fundamental mathematical principle a problem has no 
definite solution unless the number of independent equations is equal to 
the number of unknown variables. The unknown future state of the 
atmosphere is described by the velocity and the three physical variables 
of state. To solve for the vector variable v and the three scalar variables 
p, a, r, one vector equation and three scalar equations are required. 
The one vector equation is the equation of motion. The three scalar 
equations are: the equation of continuity, the equation of state, and the 
equation of energy. The solution of the general problem of atmospheric 
motion calls for the integration of these simultaneous atmospheric 
equations. 

However, the atmosphere is much too complex a " fluid system " to 
allow complete integration of the atmospheric equations with our 
present knowledge. Not only is it heterogeneous and of variable com- 
position due to condensation and evaporation, but it is also a " thermally 
active ff fluid continually receiving or losing heat. The atmospheric 
heat exchange is maintained primarily by radiation and condensation, 
which both depend upon the amount of water vapor in the air. The 
spatial distribution of the water vapor is continually being changed by 
the motion of the atmosphere. The motion, on the other hand, is pri- 
marily caused by the thermal action of heating and cooling and thus 
depends upon the moisture distribution. This interdependence between 
the motion and the distribution of water vapor leads to insurmountable 
mathematical difficulties in any complete theoretical analysis of atmos- 
pheric motion. However, some knowledge about the dynamic behav- 
ior of the atmosphere can be gained without complete integration. 

The earlier chapters in this book illustrate one type of theoretical 
approach. A number of simple dynamic rules were developed from the 
several atmospheric equations. These rules were used to study the 

337 



Section 12-01 338 

motion which is actually observed in the atmosphere. The problem was 
to understand the observed behavior of the atmosphere and explain the 
evolution of the weather in terms of these dynamic rules. The study of 
wave motion in the westerlies in chapter 10 is a good example of this 
type of mixed theoretical-empirical approach. 

If we wish to study the atmosphere with strictly theoretical methods, 
we must make some simplifying assumptions. Actually the equation 
of energy introduces an additional variable: the heat imparted to the 
moving particle. This would require further equations from the field 
of conduction and radiation of heat. It is therefore customary to elimi- 
nate the heat from dynamic problems by assuming that the changes of 
state of the air are prescribed. The temperature may then be elimi- 
nated from the equations of energy and state. The resulting equation, 
which gives the specific volume (or density) of the air particles as a func- 
tion of their pressure, is called the equation of piezotropy, and an atmos- 
phere whose physical behavior is restricted in this way is called a piezo- 
tropic atmosphere. For example, if the air particles are restricted to 
adiabatic changes the piezotropic equation is Poisson's equation. It 
should be noted that no work is performed by the air in a piezotropic 
atmosphere. The equation of piezotropy is for each particle represented 
by a line in the thermodynamic diagram. The particle can perform 
only processes along that line, and hence no cyclic process encloses any 
area. Therefore the piezotropic atmosphere, which is the only case as 
yet accessible to rigorous dynamic analysis, is thermally inactive. 

Even for the piezotropic atmosphere we find that the complete inte- 
gration of the atmospheric equations is too difficult. However, we 
already know one simple solution of the atmospheric equations, namely, 
steady zonal motion; see 11-07. Here the fields of pressure and mass 
are symmetric about the axis of the earth, and the air moves along the 
circles of latitude without any changes of state. The equations of con- 
tinuity, state, and energy are then automatically satisfied. And the 
equation of motion is satisfied when the variation of the wind in the 
direction of the axis has the value prescribed by the solenoids; see 
11*08(4). This solution also has great practical value, for it represents 
the first approximation to the motion actually observed in the atmos- 
phere, when the variations arising from the asymmetric distribution of 
continents and oceans are eliminated. The moving cyclones and anti- 
cyclones which cause the daily changes in the weather are superimposed 
on this zonal motion. We know that these disturbances have closed 
circulation only in the lowest part of the atmosphere, and that in upper 
levels they are surmounted by wave-shaped flow patterns superimposed 
on the general westerly current. We also know that these upper waves 



339 Section 12-02 

initially have small amplitudes, so that the wave motion may be regarded 
as a small perturbation superimposed on a steady zonal current. 

The first step in a rational theory of atmospheric motion is therefore 
the study of wave-shaped perturbations with small amplitudes super- 
imposed on a zonal current. In this problem the general atmospheric 
equations are reduced to linear equations which can be treated with 
relatively simple mathematical methods. V. Bjerknes has developed 
the general form of the linear atmospheric perturbation equations for 
small perturbations superimposed upon a completely arbitrary motion. 
With the aid of these perturbation equations H. Solberg and later C. L. 
Godske and B. Haurwitz have solved a large number of atmospheric 
problems in particular, problems connected with wave motion in a 
sloping frontal surface separating two zonal currents. The solutions of 
these problems have thrown much light upon the dynamics of the forma- 
tion and development of cyclones. A complete treatment of the per- 
turbation theory, and some aspects of the dynamic cyclone theory is 
given in Physikalische Hydrodynamik* chapters 7-13. 

In the present chapter we shall consider only waves in a zonal current 
without internal frontal discontinuities. The complete solution of 
waves in a barotropic zonal current is discussed in sections 12'05-12-07. 
The dynamics of wave motion in baroclinic currents is discussed quali- 
tatively later in section 12*08. 

12*02. Autobarotropy. An atmospheric current has been called 
barotropic when the surfaces of constant specific volume (the isosteric 
surfaces) coincide with the isobaric surfaces throughout the current. 
In the general baroclinic case the isosteric surfaces intersect the isobaric 
surfaces, and the current contains solenoids. 

In the barotropic case the geometric distribution of density is deter- 
mined completely by the pressure distribution. Accordingly there 
exists for each barotropic situation a relation of the form p = p(p), 
known as the equation of barotropy. For example, an atmospheric layer 
in hydrostatic equilibrium and having a constant lapse rate of virtual 
temperature is barotropic. Its equation of barotropy is given by 
448(6) when the virtual temperature is eliminated by means of the 
equation of state. 

Both barotropic and baroclinic currents may be specified to be 
piezotropic; that is to say, the physical changes of the individual moving 
particles may be prescribed by an equation of piezotropy (section 12-01). 
A current which at a given moment is barotropic does not, in general, 

* V. Bjerknes, J. Bjerknes, H. Solberg, and T. Bergeron, Physikalische Hydro- 
dynamik, Springer, Berlin, 1933. 



Section 12-02 340 

remain barotropic even if its changes are piezotropic. Assume, for 
instance, that the air within a dry atmospheric layer in hydrostatic 
equilibrium changes state adiabatically. The layer is then both baro- 
tropic and piezotropic. When this layer is disturbed, it will remain 
barotropic if the lapse rate is dry adiabatic. But for any other value of 
the lapse rate the layer will become baroclinic. 

In general a barotropic atmosphere will not remain barotropic unless 
its equation of barotropy is identical to its equation of piezotropy. If 
this condition is satisfied, the atmosphere is said to be autobarotropic. 
The corresponding equation of autobarotropy describes both the geo- 
metric distribution of the mass field in terms of the pressure field at any 
given time and also the physical change of each individual air particle 
during its motion. Simple examples of autobarotropy are: (i) an 
incompressible homogeneous atmosphere, (ii) an adiabatic atmosphere 
with adiabatic changes of state, (iii) an isothermal atmosphere with iso- 
thermal changes of state. 

The barotropic currents which are investigated in the following sec- 
tions are always assumed to be autobarotropic. For convenience they 
will nevertheless be referred to simply as barotropic currents. 

12*03. Boundary conditions. In order to obtain explicit solutions 
of the atmospheric equations boundary conditions must be introduced. 

At an internal boundary (or frontal surface) separating one air mass 
from another air mass with different motion and physical properties 
two boundary conditions must be satisfied, namely, the dynamic and the 
kinematic boundary conditions; see sections 8-08, 8-10. The solutions 
of cyclone waves in the polar front must satisfy both these boundary 
conditions. 

At an external boundary or free surface the kinematic boundary con- 
dition is automatically satisfied, for the motion of a free surface is not 
restricted. The dynamic boundary condition at a free surface requires 
simply that the pressure be zero. The only free surface of the atmos- 
phere is its outer limit. 

At a fixed rigid boundary the dynamic boundary condition is auto- 
matically satisfied, for the fixed boundary can take up any pressure 
exerted by the fluid. The kinematic boundary condition at a fixed 
boundary requires that the velocity component normal to the surface be 
zero. The only rigid boundary of the atmosphere is the surface of the 
earth. On a level part of the surface of the earth the boundary condi- 
tion is therefore that the vertical motion is zero. 

In the single layer problem to be discussed below the only two bound- 
ary conditions to be satisfied are then : (i) p = at the top of the atmos- 
phere, (ii) v g at the surface of the earth. 



341 Section 12-05 

12*04. Sinusoidal waves in a westerly current. Some of the proper- 
ties of waves in a westerly current were derived in chapter 10 with the aid 
of the transport method and the tendency equation. We shall now 
derive these results more rigorously by solving the atmospheric equations 
for small perturbations of a zonal current. We shall frequently refer to 
the earlier results during the development in the following sections. 
Whenever possible we shall use the same notations as in chapter 10. 

In chapter 10 it was unnecessary to specify the analytical expression 
for the streamlines or the isobars. The results hold on a spherical earth 
for any wave-shaped flow pattern symmetrical about the north-south 
trough and wedge lines. Only in section 10-11 was the theory specialized 
to a sinusoidal wave on a flat earth. In the following we shall restrict 
our investigation to simple harmonic waves with sinusoidal streamline 
patterns. For such waves we shall be able to undertake a more complete 
quantitative analysis than was ever possible in chapter 10, and the 
mathematical treatment can be extended to levels of diverging flow. 

Although much of the theory has been developed for a spherical earth 
by Haurwitz, we shall, in order to simplify the mathematical treatment, 
assume that the earth is flat. That is, as in 10-11, we shall consider a 
limited region of the earth where the levels may with sufficient accuracy 
be assumed to be horizontal planes. Within such a region the circles of 
latitude will be considered as parallel straight lines. 

As in chapter 10, we shall first examine waves in a barotropic westerly 
current, and later proceed to the study of waves in the more real baro- 
clinic westerly current. 

12 -OS. Waves in a barotropic current. It was shown in section 1 1 -07 
that the speed of a barotropic zonal current has no variation in the direc- 
tion parallel to the axis of the earth. The speed may have any variation 
normal to the axis. We shall now consider a barotropic westerly current 
which has no horizontal shear. Therefore in any given level the current 
has the same speed at all latitudes. It follows then that the current has 
the same speed at all levels within the region we consider. 

In this barotropic current we shall examine a sinusoidal wave disturb- 
ance which has infinite lateral extent, so that all the streamlines have the 
same- amplitude. We shall first assume that a level of non-divergence 
exists and study the motion at that level. The wave-shaped flow 
pattern at the level of non-divergence can be described by a stream func- 
tion (section 11-28). Introducing a standard Cartesian system of 
coordinates, we shall show that this stream function has the form 

(1) \l/ = v[y - AS sin k(x d)\\ 

where A $ is the streamline amplitude; k = 2w/Ls is the wave number 



Section 12-05 342 

(see 10-11); and c is the speed of the wave. The constant factor v 
must have the dimensions of a velocity to make the expression for the 
stream function dimensionally correct; see 11-28(3). It will be shown 
presently that v is the speed of the undisturbed current. 

At the time t an arbitrary streamline \l/ = const in the flow pattern 

(1) intersects the y axis at the ordinate y Q = -$/v. Substituting the 
value \l/ - -vy Q ir*(l) and dividing out the constant factor v, we find the 
equation for the streamlines: 

(2) y - y^ - AS sin k(x - ct). 

The different streamlines are obtained by assigning different values to 
y Q . They are all congruent sine curves. The streamline passing 
through the origin at the time t is the same as that examined in 
section 10-11. 

The rectangular velocity components corresponding to the stream 
function (1) are obtained from the expressions 11-28(4). We find 

(3 ) v * ~ * 

v v = vkAg cos k(x ct). 

The zonal velocity component has the constant value v throughout the 
level of non-divergence. Hence the westerly current has the same speed 
in all latitudes, in accordance with the assumption made at the outset. 

The paths of the individual air particles in the moving flow pattern .(2) 
are obtained by integration of the simultaneous system 

dx 

*-' 

(4) 

dy 

- vkAs cos k(x - ct). 

at 

The x equation can be integrated independently of the y equation. The 
air particles which are on the y axis at the time t = have at any later 
time / the abscissa 

(5) x - vt. 

Substitution of this value of x in the y equation gives dy/dt = 
vkAs cos k(v-c)t. This equation can now be integrated. The air 
particle which has the ordinate y Q on the y axis at the time / has at 
any later time t the ordinate 

(6) y yo + AS sin k(v c)t. 

The equations (5, 6) give the position of the particle with the initial 



343 Section 12-05 

coordinates (0,yo) as a function of time. When the time is eliminated, 
we have 

/*\ V A -. 

(7) y - yo = - As sin - kx. 

V C V 

This is the equation for the paths of the particles which cross the y axis 
at the time t 0. All the paths are congruent sine curves with the 
amplitude A = A$v/(v c) and the wave length L = Lgv/(v - c). 
This is in accordance with the more general formulas 11-24(3, 4). 

The vorticity of the velocity field (3) is obtained from 11-17(2). We 
have 



(8) f = * - = - vk *A s sin (* - ct). 

dx by 

The vorticity of the flow pattern (2) is thus independent of latitude. Its 
maximum cyclonic value, vk 2 A$ = vKs, occurs at the trough line, and 
its maximum anticyclonic value, vk 2 A$ = vK$, at the wedge line; 
see 10-11(6). The vorticity is zero halfway between these lines, where 
the streamline has zero curvature. 

We shall now examine what restrictions the vorticity theorem imposes 
upon the flow pattern (2). For non -diverging barotropic flow the 
vorticity theorem is 

df 212 cos <p 

11-21(6) - + - -v v =Q. 

at a 

The individual change of the vorticity may be separated into local 
and advective change by 10-04(2). Since the motion is horizontal we 
have 

1-1 i-i-l- 

When the value (8) for the vorticity is substituted in the three terms on 
the right side in (9), and the indicated differentiation is performed, we 
find 

(10) -7- - (c - v)vk*A 8 cos k(x - ct) - -k 2 (v - c)v y . 

at 

With this value for df/dt the vorticity theorem becomes 



9 / 20a cos 3 <p\ 

(11) -k 2 (v-c- - ~ 2 - j v y = 0. 

In the parentheses the angular wave number n has been substituted for 
the linear wave number k from 10-11(3). 



Section 12-05 344 

If the value of v satisfies equation (11), the stream function (1) repre- 
sents a moving wave-shaped flow pattern of such a nature that the 
individual particles retain constant absolute vorticity. Of all the flow 
patterns represented by (1), only the special one which also satisfies (11) 
has physical reality. This solution, characterized both by non-diver- 
gence and conservation of individual absolute vorticity, requires that the 
west wind relative to the moving wave have the critical speed : 

2fla cos 3 (p 

(12) v-c -- -=v c . 

n 

This result, obtained by Rossby, is approximately the same as that 
derived from the transport method; see 10-11(7) and 10-10(11). The 
earlier result holds for the level of zero longitudinal divergence, which in 
the present case of an infinitely wide wave is the level of zero horizontal 
divergence. 

Since the west wind must have the same speed at all levels in a baro- 
tropic current, equation (12) is satisfied at every level if it is satisfied at 
any one level. Therefore, as stated in section 10-14, if a level of non- 
divergence exists in a barotropic current, the entire flow is non -diverging. 
The pressure tendency is then zero at every point in the field, so the 
wave is stationary (c = 0). Therefore in a barotropic current the con- 
dition (12) is satisfied for a stationary wave only. 

Conversely, if the barotropic wave moves, the flow has horizontal 
divergence at all levels, and at no level can the velocity field be repre- 
sented by a stream function. However, if the wave has the same 
streamline amplitude in all latitudes, we shall show that the velocity 
field at a level of diverging flow is given by 

v x = v + &vsmk(x-ct), 
cos k(x - ct). 



Here v is the mean zonal wind; v+ &v and v- Av are the wind speeds 
at the wedge and trough lines, respectively. As in 1010(7), &v may 
be either positive or negative. Both v and Az> are assumed to be inde- 
pendent of latitude. The significance of the amplitude factor A so will 
be explained presently. 

Since the streamline by definition is tangent to the velocity, the 
differential equation for the streamlines is 



When the velocity components (13) are substituted in (14), this equa- 



345 Section 12-05 

tion can be integrated. The explicit equation for the streamlines of the 
velocity field (13) is found to be 

v f Az; "I 

(15) y - yo = ASO In 1 -f sin k(x - d) 

Az; L v J 

Here as in (2) yo is the ordinate where the streamline intersects the 
y axis at the time / = 0. The amplitude -4^ of the northern bend meas- 
ured from th latitude y = y Q is different from the amplitude A 3 of the 
southern bend. The two amplitudes are 



Az; 
(16) 

A s = -AsQ-^- 
Az; 

Evidently all the streamlines in the field (13) have the same amplitudes. 
If the horizontal divergence is small, \&v\ is small compared with v. We 
then have to the first order of approximation from (16) that A$ = AS 
A so, and from (IS) that 

(16') y - yo = AS sin k(x - ct). 

The streamlines are then approximately the same as the sinusoidal 
streamlines (2) at a level of non-divergence. The diverging velocity 
field (13) thus approaches the non-diverging velocity field (3) when 
A*;->0. 

The zonal velocity amplitude Az; is evidently proportional to the 
difference in transport across the trough line and the wedge line and is 
therefore a qualitative indication of the horizontal divergence in the 
current. The relation between Az; and V//*v is obtained by substitution 
of the velocity components (13) in the Cartesian expression 10-03(1) 
for Vjy'v. We then find that 



(17) V//-V - r + r - Ai* cos k(x - ct\ 

b# by 

or when v v is substituted from (13), 
(18) 



This formula shows that the divergence is zero at the trough line and 
wedge line and has a maximum or minimum value halfway between these 
lines. 
Another expression for the horizontal divergence is obtained from the 



Section 12-05 346 

vorticity theorem 11 -2 1(5): 

(19) 



7, i v y 

at a 

It is readily seen from (8) that the diverging field (13) has the same 
vorticity as the non-diverging field (3). The individual vorticity change 
in the field (13) is then given by the following equation similar to equa- 
tion (10): 

(20) f =-k 2 (v x -c)v v . 

at 

Substituting this expression in (19) and introducing the critical speed 
v c = (22 a cos 3 <p)/n 2 , we find 

(21) (f + 212JV//-V - k 2 (v x - c - v c )v y . 

This formula confirms the results obtained from the conditions 10-10(11). 
In the region to the east of the trough (v y > 0) the longitudinal diver- 
gence has the same sign as the deviation v x - c - v c of the relative zonal 
wind from the critical speed. The formula (21) gives the quantitative 
expression from which the longitudinal divergence can be evaluated when 
the deviation v x - c - v c is known. 
When the ratio is taken between (21) and (18), we find 

(f + 2^)Aw - vk 2 A S o(v x - c - v c ). 
Substituting here the value of v x from (13), we obtain 

(22) 2(f -f- 0,)Ai> = vk 2 A so (v - c - v ). 

This equation demonstrates the physical limitations of the theory pre- 
sented here. .From (8) the vorticity is given by 

(22') f - -vk 2 A SQ sin k(x - ct), 

a function of (x - ci). All the other quantities in (22) are constants 
along any given latitude. Equation (22) is then wrong, unless the 
velocity components of the wave disturbance in (13) are small. If the 
two amplitude factors &v and A$Q are small of the first order, the prod- 
uct f Av becomes smalj of the second order and may be dropped from 
(22). We then have: 

(23) 20,A - vk 2 A S o(v - c - v c ), 

which is now consistent. The simple harmonic wave motion (13) is 
therefore, strictly speaking, physically possible only when the velocity 
components due to the wave disturbance are infinitesimal. Neverthe- 
less, it is reasonable to assume that the theory holds approximately also 
for waves of not too large finite amplitude. 



347 Section 12-06 

12*06. The pressure field in the barotropic wave. When the motion 
is specified, the pressure field can be obtained from the equation of 
motion. The field of motion in the barotropic wave in the preceding 
section is given analytically in terms of the standard Cartesian com- 
ponents x,y. Accordingly, we shall use the Cartesian components of the 
equation of motion : 

7nm ^ ^ 2n 

/ lOll ) CL Za^Z/fji 

dx dt 

7-13(2) -a ^--^+ 20,1^. 

dy dt 

The individual time derivatives of the velocity components are sepa- 
rated into local and advective derivatives by 10-04(2). Since the motion 
is horizontal, we have: 

dv x dv x ^ dv x dv x dv x 

dt ~ dt Vx ~ dt V * dx Vy dy 1 

(1) 

dvy dv v dvjj dvji dvy 

ss -|- V*V^w == ~h DX ~t~ Vy * 

dt dt dt dx dy 

Non-diverging flow: We shall first examine the pressure field of the 
barotropic wave 12-05(2) at a level of non-divergence. When the 
velocity components 12-05(3) of this wave are substituted in the three 
terms on the right side in the equations (1) and differentiations are per- 
formed, we find the components of the acceleration at the level of non- 
divergence : 



, 

dt 

(2) dv 

2 _ ( v _ c)vk 2 As sin k(x - ct). 
dt 

The acceleration has no component in the # direction, so the total acceler- 
ation is everywhere directed along the y axis. 

At the inflection points of the streamlines, halfway between the 
trough line and the wedge line, the acceleration is zero. This result may 
also be anticipated from earlier knowledge, if we for a moment imagine 
the acceleration separated into tangential and normal components. A 
particle at the inflection point on a streamline is from 12-05(7) also at 
the inflection point of its path. So the curvature of the path and hence 
the normal acceleration are zero. Furthermore, v y has an extreme value 
at the inflection point. So the actual speed of the air has the maximum 



Section 12-06 



348 



value v(l + k 2 A$)$ at this point, and hence the tangential acceleration is 
zero. The flow at an inflection point on a streamline is then both geo- 
strophic and gradient. Consequently the isobars are parallel to the 
streamlines at the points of inflection on the streamline. 

At the trough and wedge lines the acceleration has its extreme values. 
At the trough line it is 



(3) 



* trough 



and at the wedge line it has the same numerical value, with negative 
sign. At every point on these lines the acceleration is directed normal 
to the path, so the tangential acceleration is zero. This also follows 
from the fact that the air speed has the minimum value v at the trough 
and wedge lines. At these lines the paths 12-05(7) have, from 10-11 (6), 
the curvature d?y/dx 2 . At the trough line the curvature is K - 
(v c)k 2 Ag/v. This verifies the last expression on the right in (3), 
the normal acceleration. Thus the tangential acceleration is zero, so the 
isobars are parallel to the streamlines at the trough and wedge lines. 
Hence these lines are also the trough and wedge lines of the pressure 
pattern. 

The acceleration field (2) thus immediately reveals several charac- 
teristics of the pressure field at the level of non-divergence. The isobars 
are in phase with the streamlines, and have the same wave length. They 




Streamline 
Isobar 



FIG. 12-06a. Isobar in sinusoidal non-diverging current. 

are parallel to the streamlines at the points of inflection on the stream- 
lines. The speed of the air has a minimum at the trough and wedge 
lines, and a maximum halfway between these lines, at the inflection 
points. Therefore air approaching the trough and wedge lines flows 
across the isobars toward higher pressure, and air leaving these lines 
flows across the isobars toward lower pressure. A pressure field which 



349 Section 12-06 

satisfies these specifications is shown qualitatively by the isobar in 
fig. 12-06a. The isobar is, of course, only roughly sinusoidal. 

The analytical expression for the pressure field is obtained when the 
acceleration components (2) and the velocity components 1205(3) are 
substituted in the component equations of motion. We find then 



(4) 



ftp 

-a ~- = -2Q 9 vkA a cos k(x - ct), 
ox 

-a - - (v - c)vk?A s sin k(x - ct) + 
fty 



We shall first integrate these equations for the special case of a homo- 
geneous and incompressible current. Later it will be shown how the 
result is modified for an arbitrary barotropic current. 

At any given time the pressure variation dp between any two neighbor- 
ing points separated by the horizontal vector element Sr is given by 

ftp ftp 
(5) Bp = V///>-5r = -^ dx + ^ dy. 

Let po denote the pressure at the origin at the time / = 0. The pressure 
at the same time at any other point (x,y) in the level is obtained by 
taking the line integral of dp along any path from the origin to the point 
(x,y). The integration is particularly simple when the path is taken 
along the y axis from the origin (0,0) to the point (Oj), and from there 
parallel to the x axis to the point (x,y). We note that dx = along the 
first part of this path (which coincides with the y axis), and that 8y = 
along the second part of the path (which is parallel to the x axis). The 
line integral of (5) from the origin to the point (x,y) is then 

(0,y) (x,y) 

f ?>P 

(6) P-PO- J f y y 

(0,0) (0,1,) 

When this equation is multiplied by the constant factor -a and the 
expressions (4) are introduced, both integrals are easily evaluated. The 
results written in the reverse order are: 



(7) a(pQ - p) = 2$l z vAs sin k(x ct) -f 2ftra (cos <p Q cos ^>), 

where <?Q and <p are the latitudes respectively of the origin and the 
ordinate y. 

The expression (7) may be checked by differentiating it partially with 
respect to x and y, and comparing the results with (4). The partial 
derivative of (7) with respect to x gives the x component of (4). But 



Section 12-06 350 

the partial derivative of (7) with respect to y is 

5/> 212 cos v> . . ,. 
a - - ik4s sin k(x ct) 



To make this expression the same as the y component of (4), we must 
have: 

/ON / M.2 212 COS p 2 

(8) (v - c)k z - = z/ c . 

a 

This condition has already been derived twice earlier, first by the trans- 
port method 10-10(11), and then in the last section 12-05(12) by the 
vorticity method. 

The condition v c - v c = at the level of non-divergence is thus 
derived here independently for the third time, and this time directly from 
the equations of motion (4). Only when the value of v satisfies this con- 
dition will the velocity field 12-05(3) represent a solution of the equation 
of motion. Only then will the velocity field be adjusted to a pressure 
field (7) which is dynamically possible and consistent with the motion. 

When the condition v - c - v c = is satisfied, equation (7) gives the 
analytical expression for the pressure field in a homogeneous wave. For 
an arbitrary barotropic wave the development is similar. The equation 
of autobarotropy gives a as a function of p. We may therefore define a 
function TT of p as follows: 



(9) 



\CLt 



The function ir is called the barotropic pressure function. It is equal to 
the dynamic elevation of the top of the atmosphere above the point 
with the pressure p. We have STT - a8p, and therefore dir/dx = adp/dx, 
d7r/dy = abpfoy. When these expressions are substituted in (4) the 
integration may be carried out exactly as in the homogeneous case, and 
we find the following equation similar to equation (7) : 

(10) VQ - IT - -2Q, z vAs sin k(x - ct) + 212m (cos <p Q - cos <p). 

Here TT O = ic(po) is the value at the origin at the time / 0. An isobar, 
9T - const, of the barotropic field (10) has the same shape as the corre- 
sponding isobar in the homogeneous field (7). Only the spacing of the 
isobars is different. 

Consider the isobar PQ passing through the origin at the time t 0; 
from (7) it has the equation: 

2!2a (cos <p - cos <?$) -212^5 sin k(x - ct). 



351 Section 12-06 

Introducing A $ = a tan <TS and dividing out the factor 2fia, we find 
(11) cos <p cos <PQ = tan <TS sin <p sin k(x ct). 

Since the origin may be placed in any latitude, this equation holds for 
any isobar in the field. Equation (11) confirms the earlier results as to 
the relation between the pressure pattern and the flow pattern: The 
two patterns are in phase and have the same wave length. The equa- 
tion shows further that the shape of the isobar is independent of v and c. 
Therefore, the shape of the pressure pattern is completely determined 
by the shape of the flow patterns, and it is independent of the propaga- 
tion of the flow pattern and the intensity of the flow. 

To investigate how much the pressure pattern deviates from the flow 
pattern, we shall determine the latitudes of the southern and northern 
bends of the isobar (11). As in 1(MO, we denote these latitudes respec- 
tively by <p and ^/. At the two bends we have then from (11): 

cos (p cos tf>n = tan <TS sin ^>, 

/ . / 

cos <p cos <PQ = tan S sm <p . 

Subtracting the lower from the upper equation, and introducing the 
notation of 10-10(4), we find: 

2 sin sin <r p = 2 tan 0$ sin <p cos <r p , 
or 

(13) tan op = tan <?$ 

The isobars and the streamlines have the same amplitudes at a level of 
non-divergence. 

If the wave is stationary this result follows immediately from a funda- 
mental theorem by Bernoulli. This theorem is most conveniently 
derived from the tangential equation of motion: Using the formula 
10-04(2), the tangential acceleration is expressed as the sum of the local 
and the advective changes: 

dv dv 



In steady flow no local changes occur, so dv/bt - 0. The component of 
Vfl along v is cto/ds, so the advective change is v*V - vdv/ds. We sub- 
stitute this value of the tangential acceleration in 7*13(4) and find 

dv dp 

0--= -a - 

5s bs 

Introducing from (9) the barotropic pressure function IT we have 



Section 12-06 352 

adp/ds - d7r/ds. The above equation then becomes : 

(14) s(l + 

This is Bernoulli's theorem in differential form. It may be stated as 
follows: When a barotropic fluid has any steady horizontal motion the sum 
of the kinetic energy per unit mass and the barotropic pressure function has 
no variation along the streamlines. The theorem is named after Daniel 
Bernoulli (1700-1783), who founded the science of hydrodynamics. 

We shall now apply Bernoulli's theorem to the stationary barotropic 
wave. Since the speed is the same at the southern and northern bends 
of the streamline, the barotropic pressure function also has the same 
value at these points. So the same isobar passes through both bends 
of the streamline and has accordingly the streamline amplitude. Ber- 
noulli's theorem also shows that the streamlines have their highest 
pressure at the trough and wedge where the speed is minimum, and their 
lowest pressure at the inflection points where the speed is maximum; 
see fig. 12-06a. The pressure drop along the streamline from trough (or 
wedge) to inflection point is given by : 



For example, at a level where v = 10 ms" 1 , a = 1000 m 3 IT 1 , a wave at 
60 lat having n = 6, 0-5 = 5 would have a pressure variation A = O.S 
mb along its streamlines. The deviation of the isobar from the stream- 
line in a stationary wave may be estimated by this method. 

The deviation of the pressure pattern from the flow pattern is obtained 
more directly from the equations (12), which hold for both a moving and 
a stationary wave. When these equations are added we find 

2 cos ^ cos <7 P - 2 cos y? = -2 tan 0-5 cos sin <r p . 

We substitute here from (13) as = * P > and find after rearrangement of 
the terms: 

(15) , COS COS <?Q COS <7 p . 

The central latitude of the isobar is accordingly north of the latitude 
<f> Q , as indicated in fig. 12-Ofo. Equation (IS) holds for any isobar in the 
field. The isobar passing through the inflection points of an arbitrary 
streamline deviates at the trough and wedge from that streamline by the 
amount ^ - <?Q. The values of - y? f r different latitudes and ampli- 
tudes are shown in table 12-060, computed from (15). The value of 
^ <?$ is a good indication of the deviation of the pressure pattern from 
the flow pattern. In a given latitude the deviation is determined com- 



353 Section 12-06 

pletely by the amplitude of the isobar and is independent of the wave 
length, the wind speed, and the speed of the wave. In high latitudes the 
deviation of the two patterns is negligible. In low latitudes the devia- 
tion is appreciable, particularly when the amplitude is large. 



TABLE 12.06a 
<PQ TABULATED 




<f>0 


Op 


10 


30 


50 


70 


5 


1.2 


0.4 


0.2 


0.1 


10 


4.1 


1.5 


0.7 


0.3 


15 





3.2 


1.6 


0.7 



It was shown in section 10-14 that the non -diverging barotropic wave 
must be stationary. It may therefore seem unnecessary to investigate 
its pressure field as if it were a moving wave. However, the results 
derived above will hold approximately at a level of non-divergence in a 
baroclinic wave. We shall therefore need the general derivation for 
moving waves later in the study of the baroclinic waves. 

Diverging flow: We shall now examine the pressure field in the moving 
barotropic wave, which has horizontal divergence at all levels. It has 
been shown that the velocity field at any level in such a wave is given by 
12-05(13) when the two amplitude factors |Ay| and A so are sufficiently 
small. For small |AI>| the amplitude factor A so becomes the streamline 
amplitude A s ; see 12-05(16). In the following development we shall 
accordingly write A so = AS and drop all terms containing squares or 
products of Az; and A s. The streamlines of the wave 12-05 (13) are given 
by 12-05(15). For infinitesimal amplitudes this expression becomes 

(16) y - y Q = AS sin k(x - ct). 

The flow pattern has simple sinusoidal streamlines as in the case of non- 
diverging flow. 

The pressure field is obtained from the equations of motion. We 
first substitute the velocity field 12-05(13) in (1) and find the corre- 
sponding acceleration field: 

- = (D c) &vk cos k{x ct), 
at 

(17) * 

2 = _ ( _ c)vk 2 As sin k(x ct). 
at 



Section 12-06 354 

These expressions and the velocity components are introduced in the 
component equations of motion, which then become: 

d 

-a =* \(v - c)Av - 2Sl z vAs\k cos k(x - ct), 

ox 
(18) 

-a:~ = -[(t> - c)D^ 2 yl 5 - 2Q,At>] sin *(* - d) + 20,0. 
d^ 

These equations may now be integrated exactly like the equations (4), 
and we find an equation similar to (10) : 



(19) TT O - TT = [(v - c)Av -2QJ)As] sin k(x - ct) + 2fiat> (cos^ - cos^), 

where ir is given by (9). The partial derivative of (19) with respect to 
x gives the x equation (18). The partial derivative of (19) with respect 
to y is the same as the y equation (18) only when 



(20) 2U z &v - vk 2 A s (v -c)-QA s = vk 2 A s (v -c- v c ). 

a 

But this is the earlier condition 12-05(23), derived from the vorticity 
theorem. Only when the parameters in the velocity field 12-05(13) 
satisfy this condition does that field represent a solution of the equation 
of motion. 

We substitute in (19) the value of Az; from (20) and introduce two 
abbreviations: 

k n 

2Q 2 "~ 120 sin 2<p* 

(22) 77f 2 (v-c)(y~c-t; c ). 
The resulting equation 

(23) TT O - TT = -20,8(1 - r})A s sin k(x - ct) + 2tiav (cos ^ - cos ^) 

gives the barotropic pressure function in the wave with the velocity field 
12'05(13). We shall now study the relation between the flow pattern 
(16) and the pressure field (23) in this wave. 
For the isobar TT O through the origin at t = we find, similar to (11), 

(24) cos <p cos <?Q = - (1 77) tan cr s sin v sin k(x ct). 

This equation differs from (11) only in the amplitude factor (1 TJ). 
When the development subsequent to (11) is repeated for (24), we find 

(25) tan <r p = (1 - rj) tan cr^, 

(26) COS <p = COS <{>Q COS ff p . 



355 Section 12-06 

The deviation - VQ between the central latitudes of a streamline and 
the isobar passing through its inflection points is from (26) and (15) the 
same in the diverging and the non-diverging wave. 

The amplitudes are equal when rj = 0, that is, when the relative zonal 
wind either is zero or has the critical speed v c . In the first case both 
amplitudes are zero; see 10*15(3). In the second case the flow is non- 
diverging and (25) reduces to (13). For supercritical and for negative 
relative zonal wind (77 > 0) the streamline amplitude is larger than the 
isobar amplitude. For subcritical, positive relative zonal wind (rj < 0) 
the streamline amplitude is smaller than the isobar amplitude; see 
fig. 12-066. 

For any given value of the relative wind the magnitude of the ampli- 
tude difference is determined by the factor 2 in 17. The values of for 
different latitudes and wave numbers are shown in table 12-066. For a 

TABLE 12-066 
= n(Qa sin 2^)~ 1 M" 1 s, TABULATED 





n 


<p 


2 


3 


6 


10 


20 


45 


0.0043 


0.0064 


0.0129 


0.0216 


0.0431 


40, 50 


0.0044 


0.0066 


0.0132 


0.0219 


0.0438 


30, 60 


0.0050 


0.0075 


0.0149 


0.0249 


0.0498 


20, 70 


0.0067 


0.0101 


0.0201 


0.0335 


0.0671 


10, 80 


0.0126 


0.0189 


0.0378 


0.0630 


0.1260 



given wave length the minimum value of occurs at 45 lat and its varia- 
tion in the belt from 30 to 60 lat is quite small. To estimate the value 
of 77 in this range of latitudes, consider a case where the relative zonal 
wind and its deviation from the critical speed both have the order of 
magnitude 10m s" 1 . For a relatively long wave (n = 6) rj is then about 
0.02. For a short wave (n 20) r; is about 0.2. We shall show in the 
next section that rf is usually very small in the barotropic wave. So the 
amplitude difference is therefore negligible. However, the above analy- 
sis also holds approximately at a level of diverging flow in a baroclinic 
wave. Since the wind increases with height in these waves, both the 
relative wind and its deviation from the critical value may be large at 
high levels. Here TJ, and hence the amplitude difference, may be appre- 
ciable, particularly for short waves in very high or very low latitudes. 

It remains to show how the isobar runs at the inflection points of the 
streamline. Let if/s and \l/ p denote the angles which the streamline and 
the isobar make with the x axis at these points. Differentiation of (16) 



Section 12-06 356 

and (24) gives at the inflection point east of the trough: 



therefore 

(27) tan^ p = (1-tj) tantf^. 

Accordingly, the angle \l/$ ^ p has the same sign as rj. For non- 
diverging flow (77 = 0) the isobar is parallel to the streamline at the 
inflection point in accordance with the earlier result. For supercritical 
and for negative relative zonal wind (77 > 0) the air at the inflection point 




Streamline 
} isobars 



FIG. 12-066. Isobars in sinusoidal diverging current. 

east of the trough flows across the isobar toward higher pressure. And 
for subcritical, positive relative zonal wind (rj < 0) the air at that point 
flows across the isobars toward lower pressure. The three cases are 
shown schematically in fig. 12-06&. Since t? is very small in the baro- 
tropic wave, the isobars are here nearly parallel to the streamlines at the 
inflection point. 

12*07. The speed of propagation of the barotropic wave. In the 
analysis in the preceding section not all the atmospheric equations at our 
disposal have been used. The equations of state and of thermal energy 
are automatically accounted for by the equation of autobarotropy. 
Besides this condition only the horizontal component of the equation of 
motion has been considered so far. The remaining equations are the 
vertical (hydrostatic) equation and the equation of continuity. These 
equations enable us to find the speed of propagation of the barotropic 
wave. 



357 Section 12-07 

It was shown in section 10-07 how the hydrostatic equation and the 
equation of continuity combine into the tendency equation. We shall 
first evaluate the pressure tendency at the surface of the earth. The 
tendency equation is here 

oo 

10-07(4) f-. 



Note that by writing the tendency equation in this form we make use of 
the boundary condition v z = at the lower boundary of the atmosphere. 
We separate the mass divergence into two terms, as follows: 

V//.(pv) - pV//*v + vV//p. 

It was stated in the preceding section that the isobars, and hence the 
isosteres, are very nearly parallel to the streamlines in the barotropic 
wave, so vV//p 0. The justification of this approximation is dis- 
cussed at the end of this section. Substituting from 12-05(17) for the 
horizontal velocity divergence we find then 

V#*(pv) = p&vk cos k(x ct). 

The value of &v is given by 12-06(20). It was shown at the outset in 
section 12-05 that the barotropic current has the same speed at all 
levels. That makes &v also the same at all heights. The tendency at 
the surface is therefore 



(1) = &vk cos k(x - ct) I p8<t>. 

o 

From the hydrostatic equation we have pd<t> = -5/>. Hence the integral 
in (1) is equal to the pressure, />, at the ground. Note that we here 
make use of the boundary condition p = at the upper boundary of the 
atmosphere. 

We multiply equation (1) by -a, introduce from 4-19(5) ap** 
R d T* = fa, and from 12-06(9) the barotropic pressure function TT. 
We then get 

(2) - = fa&vk cos k(x - ct), 

Qt 

where fa is the dynamic height of the homogeneous atmosphere. 

Partial differentiation of 12-06(23) with respect to time gives on the 
other hand 



(3) - = 2Q 2 vAsc(l ~ rj)k cos k(x - ct). 



Section 12-07 358 

Since the two expressions (2, 3) must be the same, we have the condition: 

(4) <feAf - 2& z vA s c(\ - r;). 
Earlier we derived the condition 

12-06(20) 2Q z Av k 2 vA s (v -c- v c ). 

Only when the parameters in the velocity field 12-05 (13) satisfy both 
these conditions does that field represent a solution of the complete set 
of atmospheric equations for a barotropic wave. 

We now take the ratio of the two dynamical conditions above and 
introduce from 12-06(21). The resulting equation is 

(5) ^(tJ-c-O-ctt-*). 

Note that the complete set of atmospheric equations and also the 
boundary conditions at the bottom and at the top of the atmosphere 
were needed to derive this equation. Equation (5) then represents the 
complete solution for a wave in a barotropic zonal current. It gives c 
implicity as a function of latitude, wave number, and zonal wind speed. 
However, since t\ contains c 2 , (5) is a cubic equation in c and therefore 
rather difficult to discuss in its present form. To facilitate the dis- 
cussion we transform it as follows: First we multiply (5) by (v - c) 
and introduce rj from 12-06(22). We then get 



or 



We now substitute again for r\ and divide out the factor (v c). The 
resulting equation is: 

(6) c - ?(<i> h + c(v - c)](v - c - v ) (exact). 

The numerical value of <t>h = R&T* is about 280 2 mV~ 2 , whereas c(v c) 
certainly must be less than say 28 2 m 2 s"~ 2 . The value of c with an error 
of less than 1% is therefore 

(7) c~?<t> h (v-c-v c ). 

This equation confirms the qualitative rules derived earlier in sec- 
tion 10-14. The wave is stationary (c = 0) when the zonal wind has the 
critical speed. The wave moves toward the east (c > 0) when the zonal 
wind is supercritical and toward the west (c < 0) when the zonal wind is 
subcritical. Equation (7) shows further that c is proportional to 
v - c - v c . From 12*05(18, 23), that means that the speed of the wave is 
proportional to the horizontal divergence in the current, which is just 
what the tendency equation predicts in a qualitative sense. 



359 Section 12-07 

Solving finally for c from (7), we have 

2 </> 

(8) c = (v - v c ) ~ (first approximation). 

1 + <t>h 

Rossby (1939) derived essentially the same formula as (8). The 
mts values of for different latitudes and wave numbers are given in 
table 12-06&. Corresponding values of the non-dimensional factor in 
(8) are given in table 12-07. The table has been computed for <fo = 

TABLE 12-07 

+ ^h)" 1 TABULATED 





n 


^ 


2 


3 


6 


10 


20 


45 


0.59 


0.76 


0.93 


0.97 


0.99 


40, 50 


0.60 


0.77 


0.93 


0.97 


0.99 


30, 60 


0.66 


0.81 


0.94 


0.98 


0.99 


20, 70 


0.78 


0.89 


0.97 


0.99 


1.00 


10, 80 


0.93 


0.97 


0.99 


1.00 


1.00 



280 2 = 78,400 m 2 s 2 , which corresponds to a surface temperature of 
about 273K. However, the values in the table can be used safely for 
any observed surface temperature. The table shows that the non- 
dimensional factor in (8) is considerably smaller than 1 for long waves in 
middle latitudes, but the factor rapidly approaches 1 for shorter waves. 
For waves of the same order of magnitude as those actually observed in 
the atmosphere (wave lengths 60 of longitude or less) the wave moves 
approximately in accordance with the simple formula 

c = v v c (second approximation). 

The barotropic waves move then approximately as though the flow were 
non-diverging. Physically, this means that the deviation from non- 
divergence required to move the barotropic waves along with the speed c 
is a small fraction of c if the wave is short but is a considerable fraction 
of c for very long waves. The values of v c were given in table 10-11. 

It remains to find the value of rj for the barotropic wave. We substi- 
tute in (5) for c from (6) and divide out the factor 2 (t> - c - v c ). We 
find then 



or, when we solve for 

m 



>- c) 



Section 12-07 360 

Since c(v - c) usually is less than 1% of <fo, r? will be smaller than 0.01. 
Therefore the isobars have practically the same amplitudes as the stream- 
lines and they are very nearly parallel to the streamlines at the inflection 
points. This makes permissible the neglect of the horizontal density 
advection v*V#P in the tendency equation, which is the only physical 
approximation made in this section. 

12-08. Waves in a baroclinic current. The atmosphere is always 
more or less baroclinic, with cold and heavy masses in the polar regions, 
and with warm and light masses in the equatorial region. The isobaric 
layers are accordingly inflated in the direction from the poles to the 
equator. In the case of zonal symmetry the undisturbed zonal current 
increases in intensity with increasing distance from the equatorial 
plane in accordance with the exact formula: 

(dv\ 1 /da\ 
^i) - -(T) 
b<t>/ R a\5s/ p 

We shall consider only the belt of latitudes which has west winds at the 
surface of the earth. Within this belt the west wind increases in all 
latitudes with increasing distance from the equatorial plane. In the 
direction normal to the axis the speed may have any variation. To 
simplify matters we shall assume that the wind is constant throughout 
each horizontal level. The vertical shear of the wind, 5v/50, is then also 
constant throughout each level. Since the isobaric surfaces are nearly 
horizontal we have further co a sin <p p Q, z . The above formula becomes 
then approximately : 



- - 

d</> 2B, T dy 2Qa d(cos^) 

which is identical to the thermal wind formula for the geostrophic wind. 
The assumption of a constant wind throughout each level therefore pre- 
scribes the variation of the temperature with latitude to be as follows: 

In T - In T P + p cos ^. 

T P is the temperature at the pole and = 212adt>/d<. 

If a harmonic wave disturbance is superimposed on this baroclinic 
westerly current, the resulting velocity field at an arbitrary level is given 
by 

v x **v+&o sin k(x ct) t 

v y = vkAs cos k(x - ct) . 

The symbols have the same meaning as in 12-05(13). For small |Air| 
we have shown from 12-05(16) that A $ is the streamline amplitude. 



361 Section 12-08 

The streamlines of the velocity field (1) are, from 12-05(16'), 

(2) y yo = .4s sin k(x ct). 

Just as in 12 -05 (22'), we find that the velocity field (1) has the vorticity 

(3) f -vk 2 A s sin k(x - ct). 

And, just as in 12-05(17), we find that the velocity field (1) has the 
divergence 

(4) V//-V = Avk cos k(x - ct). 

If the motion (1) shall be dynamically possible, the expressions (3, 4) 
for the vorticity and the divergence must satisfy the vorticity theorem 

11-21(5): 



(5) 

at a 

It should be noted that this theorem is exact only when no solenoids 
intersect the horizontal level; see 11-20. In the undisturbed baroclinic 
current the solenoids are directed along the latitude circles and do not 
intersect the levels. When the baroclinic current is disturbed by the 
wave perturbation the field of solenoids is modified too. But the sole- 
noids will remain approximately horizontal and very few will intersect the 
levels. The vorticity theorem (5) therefore applies, in the first approxi- 
mation, to the velocity field (1) of the baroclinic wave. When the values 
(3, 4) of the vorticity and the divergence are substituted in (5), and 
higher order terms are neglected, we find that: 

(6) 2Q z Av = vk 2 A s (v-c- v c ). 

When Az; satisfies this equation, the velocity field (1) represents a flow 
pattern which is physically possible at all levels. The critical speed v c 
(see table 10-11) is determined by the wave length and the latitude and 
has the same value at all levels. The strength of the relative zonal 
circulation, v c, increases with height in the baroclinic westerlies. If it 
is assumed that the wave moves with a slightly lower speed than the air 
at the ground, the relative zonal wind increases from a small positive 
value at the ground to a maximum value at the tropopause. The devia- 
tion A0 is zero and the flow is non-diverging at the level where the rela- 
tive zonal wind has the critical value, v - c - v c = 0. Above the level of 
non-divergence the relative zonal wind is supercritical, and At; > 0. 
The bottleneck of the flow is then at the cyclonic trough, which there- 
fore has divergence to its east. Between the level of non-divergence 
and the ground the relative zonal wind is subcritical, and Av < 0. The 



Section 12-08 



362 



bottleneck of the flow is here at the anticyclonic wedge, and the flow has 
convergence to the east of the trough. 

The pressure field in the wave with the velocity field (1) is obtained 
by the procedure outlined in section 12-06. Just as in 12*06(24), we find 
for the isobar through the origin at the time / = the equation : 

(7) cos <p cos <PQ = (1 i?) tan <?$ sin <p sin k(x ct), 

where 

(8) -n=?(v-c)(v-c-v c ). 

The relation between the amplitudes of the streamline and the isobar is: 

(9) tan ff p = (1 - r?) tan S . 

The relation between the central latitude <p Q of the streamline and the 
central latitude <p of the isobar passing through its inflection points is: 

(10) cos <p = cos <PQ cos > 

The deviation between the isobars and the streamlines at the inflection 
points is given by 12-06(27) : 

(11) tan \l/ p = (1 - rj) tan \l/ s . 

The relation between the pressure pattern and the flow pattern at the 
various levels of the baroclinic wave is readily obtained from the three 
formulas (9, 10, 11). The deviation between the two patterns depends 

I 




10 20 0.1 

v * 17 * 

FIG. 12-08. 

upon the value of 17. From (8) rj is zero at the level where the relative 
zonal wind is zero and at the level of non-divergence, where the relative 
zonal wind has the critical speed. At these levels the isobars are tan- 
gential to the streamlines at the points halfway between the troughs 



363 Section 12-08 

and the wedges, and the two patterns have exactly the same amplitudes. 

Above the level of non-divergence and below the level where the rela- 
tive zonal wind is zero rj is positive. And in the layer between those 
two levels 77 is negative; fig. 12-08. The corresponding deviation of the 
isobars from the streamlines is as indicated schematically in fig. 12 '066. 

In the barotropic waves the relative zonal wind never deviates much 
from the critical speed. So 77 is always small, and the pressure pattern 
coincides approximately with the flow pattern at all levels in the baro- 
tropic waves. In the baroclinic waves, on the other hand, the relative 
zonal wind may be considerably greater than the critical speed at levels 
high above the level of non -divergence; see fig. 12-08. At these levels 
the magnitude of 17 becomes appreciable, and therefore the departure of 
the pressure pattern from the flow pattern is no longer negligible. As a 
general rule the deviation between the two patterns is most pronounced 
at high levels when the level of non-divergence is low and the current is 
strongly baroclinic. 

The above remarks throw some light upon the dynamics of the baro- 
clinic waves and on their three-dimensional structure. The principal 
problem remaining to be solved for these waves is the determination of 
their velocity of propagation. To solve that problem the tendency 
equation would have to be integrated for the baroclinic wave. The 
integration was rather simple for the barotropic wave, because the 
horizontal divergence is independent of height and the flow patterns at 
all levels are in phase. In the baroclinic waves not only does the hori- 
zontal divergence vary with height, but the troughs and wedges will in 
general tilt with height because of the asymmetry between the tempera- 
ture field and the pressure field; see fig. 10*17d. The points along 
a vertical column will therefore have different positions relative to the 
flow pattern at different levels. This makes the integration of the 
tendency equation for the baroclinic waves very difficult. When we 
have successfully performed the integration, the velocity of propagation 
of the baroclinic wave can be determined, and the conditions for dynami- 
cal instability of the waves, as derived by qualitative reasoning in sec- 
tion 10*17, can be obtained. 



INDEX 



Abnormal flow, 162, 202, 204 
Absolute acceleration, 170 
Absolute circulation, 313 
Absolute circulation theorem, 313 
Absolute frame, 152 
Absolute instability, 131 
Absolute motion, 151 

equation of, 154 
Absolute stability, 130 
Absolute temperature, 1 1 , 34 
Absolute velocity, 168 
Absolute vorticity, 322 

theorem of, 323 
Acceleration, 3, 148 

absolute and relative, 170 

centripetal, 150 

in horizontal path, 183 

in vertical path, 183 

in zonal flow, 159, 171 

of a point of the earth, 156, 168, 170 

Coriolis, 170 

dimensions of, 3 

in horizontal path, 183 

in vertical path, 183 

in zonal flow, 159, 171 

natural components of, 151, 175 

of gravity, 88 

of a point of the earth, 156, 168, 170 

tangential, 150, 175, 189 

units of, 6 

Adiabatic heating, 68 
Adiabatic process, 27, 64, 67 

of dry air, 28 

of unsaturated moist air, 64 

pseudo-, 68, 69 

reversible saturation-, 68, 70 

saturation-, 72 
Adiabats, 28 

dry, 29 

on emagram, 36, 37 

on Stiive diagram, 30 

on tephigram, 39 

saturation, 72 

unsaturated, 65, 139 



Advection, of density, 259 

of temperature, 220, 279, 285 
Advective pressure tendency, 258 
Advective rate of change, 254 
Air, composition of, 16 

dry, see Dry air 

moist, see Moist air 

saturated, 62 

unsaturated, 62, 65 
(a,-/0-diagram, 18, 29, 300 
Altimeter (pressure), 121 
Altimeter errors, 122 
Altimeter setting, 122 
Altitude, sea level, 121 

standard, 119 
Amplitude, angular, 266 
of isobar, 266, 351, 354 
of path, 331 
of streamline, 269, 331, 351, 354 

of path, 330, 343 

of relative streamline, 277, 330 

of streamline, 269, 330, 345 
Anemometer level, 241, 244 
Angular amplitude, see Amplitude, 

angular 
Angular radius of curvature, 177, 183, 

289 
Angular speed, 151 

in inert ial flow, 196 

in zonal flow, 160 

of the earth, 155, 177 
Angular velocity, 3, 163 

of the earth, 163, 183 
components of, 183 
vertical component of, 184, 191 

vertical component of, 180 
Angular wave number, 269 
Antibaric flow, 190, 204 
Anticyclone, abnormal, 162, 202, 204 

of maximum strength, 162, 201 

structure of, 294 

Anticyclonic circulation, 317, 318 
Anticyclonic flow, 191, 204, 207 

zonal, 160 



365 



INDEX 



366 



Anticyclonic sense, 191, 317, 323 

Anticyclonic vorticity, 323 

Apparent gravity, 227 

Apparent latitude, 229 

Apparent level, 227 

Arc length, 149 

Area, 2 ^ 

on thermodynamic diagrams, 19, 35, 
37, 39, 109, 133, 301 

under hodograph, 260 

vector, 260, 315, 316 
Area computer for advective pressure 

change, 262 
Ascendent, 97 

isobaric volume, 213 

volume, 301 
Atmosphere, baroclinic, 102 

barotropic, 102, 340 

composition of, 16 

dry-acliabatic, 106 

homogeneous, 102, 105 

isothermal, 107 

normal, 8 

U. S. standard, 119 

with constant lapse rate, 104 
Atmospheric column, weight of, 103, 

286 

Atmospheric equations, 337 
Atmospheric process, 27, 337 
Autobarotropy, 313, 339 
Avogadro's law, 14 

Bar, 6 

Baric flow, 190, 201,204 
Baric wind law, 192 
Baroclinic atmosphere, 102 
Baroclinic current, 286 
Baroclinic wave, 279, 360 

dynamic instability of, 286 

formation of, 281 

mass divergence in, 280, 285 

pressure changes in, 281 

speed of, 274 

stable, 274 
Barometer, 7, 121 

Barometric height formula, 103, 297 
Barotropic atmosphere, 102, 340 
Barotropic current, 341 
Barotropic layer, 211 
Barotropic pressure function, 350 



Barotropic wave, 274, 341 

mass divergence in, 274 

pressure changes in, 274, 276, 357 

pressure field in, 347 

speed of, 276, 356 
Barye, 5 
Base level, 120 

Bergeron's theory of precipitation, 55 
Bernoulli's theorem, 352 
Bjerknes, J., convection theory of, 135 

cyclone theory of, 256, 281 
Bjerknes, V., circulation theorem of, 299, 
302, 313, 317 

hydrostatic tables of, 111 
Boiling point, normal, 50, 51 
Boiling temperature, 50 
Boundary conditions, 221, 225, 244, 340, 
357 

dynamic, 221, 340 

kinematic, 225, 340 
Boyle's law, 12 
Buys Ballot's law, 192 

Calorie, 21 

Calorimeter, 20 

Cartesian (rectangular) components of a 

vector, 85, 174 
Cartesian coordinates, 83 
Center of curvature, 150 
Centibar, 5, 7 
Centigrade, 11 
Centrifugal force of a point of the earth, 

89, 156 

Centrifugal potential, 157 
Centripetal acceleration, 150 

in horizontal path, 183 

in vertical path, 183 

in zonal flow, 159, 171 

of a point of the earth, 156, 168, 170 
Cgs units, 5, 6 
Channel, isobaric, 263 

isobaric unit, 100 
Characteristic curve, 135 
Characteristic point, 67, 76, 133 
Charles's law, 12 
Circle, curvature of, 149 

great, 175, 180, 181, 182, 233 

inertial, 198 

of curvature, 150 

of horizontal curvature, 180 



367 



INDEX 



Circle, small, 175 

zonal, 156 

Circular frequency, 131 
Circular vortex, 308 

local, 319 
Circulation, 296 

absolute, 313 

anticyclonic, 317, 318 

cyclic sense of, 317 

cyclonic, 317, 318 

in meridional plane, 309 

in zonal flow, 314 

rate of change of, 313, 317 

relative, 315, 317 
Circulation integral, 296 
Circulation theorem, 317 

absolute, 313 

individual, 311 

of Bjcrknes, 299, 302, 313, 317 

of Kelvin, 313 

primitive, 298 

relative, 317 

Clapeyron's equation, 49, 50-1 
Closed flow patterns, 264, 288, 294 

mass divergence in, 291 
Column, atmospheric, weight of, 103, 

286 
Component equations of relative motion, 

186 
Components of vector, 85 

natural, 151, 174 

rectangular, 85, 174 
Composition of dry air, 16 
Condensation, 41, 68, 72, 77 
Condensation level, lifting, 134 
Conditional instability, 131 
Conservation, of energy, 21, 91 

of mass, 250 

Continuity, equation of, 250, 252, 337 
Convection, cumulus, 136 

level of free, 134 
Convergence, see Divergence 
Cooling, adiabatic, 68 

isobaric, 72 

super-, 55 
Coordinates, local, 174 

natural, 151, 173 

rectangular or Cartesian, 83 

standard, 83 
Coriolis acceleration, 170 



Coriolis force, 172, 184 

components of, 184 

horizontal, 185 

Corresponding curvatures, 190 
Corresponding latitudes, 190 
Corresponding normal pressure forces, 

190 

Critical constants, for permanent gases, 
43 

for water substance, 42 
Critical eccentricity, 290 
Critical lapse rate, 138 
Critical point, 50, 51 
Critical speed, 268, 271, 344 
Critical state, 42 
Critical temperature, 41 
Cumulus convection, 136 
Curvature, 149 

angular radius of, 177, 183, 289 

center of, 1 50 

circle of horizontal, 180 

corresponding, 190 

cyclic sense of, 176, 190 

geodesic, 181 

horizontal, 178, 182, 265 

of streamline and path, 207, 267 

of circle, 149 

of sinusoidal streamline, 270 

radius of, 150 

radius of horizontal, 180 

vector, 151 

vertical, 182 
Cyclic sense, 176 

of circulation, 317 

of curvature, 176, 190 

of the earth's rotation, 177 

of vorticity, 323 
Cyclone, structure of, 292 
Cyclonic circulation, 317, 318 
Cyclonic flow, 190, 204, 207 

zonal, 160 

Cyclonic sense, 191, 317, 323 
Cyclonic vorticity, 323 
Cyclostrophic flow, 199 
Cyclostrophic speed, 199, 203 

Dalton's law, 15 
Day, sidereal, 155 

solar, 155 
Decimeter, dynamic, 94 



INDEX 



368 



Degree, absolute, (K), 11 
centigrade (C), 11 
Fahrenheit (F), 12 
of latitude, 5, 192 
Del (V), 95, 251 
Density, 3, 9 
local change of, 253 
of water, 5, 44 
Density advection, 259 
Derivative, individual, 146, 253 
local (time-), 253 
of a vector, 146, 165, 251, 312 
Deviation, gcoslrophic, 234, 244 
Dew point temperature, 77 
Diagrams, thermodynamic, 34 
(,-/>) -diagram, 18, 29, 300 
emagram, 35 
height evaluation on, 115 
important criteria of, 35 
precipitation lines on, 143 
saturation adiabats on, 74 
Stiive, 30, 35 
(J^-diagram, 49 
tephigram, 38, 74 
vapor lines on, 61, 74 
Differentials, exact, 32, 92 
geometric (5), 92, 126 
individual (</), 92, 125, 146 
inexact, 32 
process (</), 92, 125 
Differentiation, logarithmic, 26 
of a vector, 146, 165, 312 

divergence-, 251 
Dimensions, 2 
Distributive law, for scalar product, 87 

for vector product, 165 
Divergence, 251 
frictional, 263 
horizontal, 253, 324, 333 
mass, 251 

horizontal, 256 
in baroclinic wave, 280, 285 
in barotropic wave, 274 
in closed flow patterns, 291 
in tendency equation, 257 
longitudinal, 263, 268, 331 
transversal, 264, 271 
Dry-adiabatic atmosphere, 107 
Dry-adiabatic lapse rate, 107, 129 
Dry-adiabatic layer, height of, 110 



Dry-adiabatic process, 28 

Dry adiabats, 29 

Dry air, composition of, 16 

equation of state of, 13 

molecular weight of, 17 

Poisson's constants for, 28, 31 

specific gas constant of, 14, 16 

specific heats of, 25 
Dry stage, 62 
Dynamic boundary condition, 221, 

340 

Dynamic decimeter, 94 
Dynamic height, 94 

computation of, 114, 115, 117 
by dry-adiabatic layer, 110 
by isothermal layer, 111 

on emagram, 109 
Dynamic instability, 286 
Dynamic meter, 95 
Dyne, 5 

Earth, acceleration of a point of, 156, 
168, 170 

angular speed of, 155, 177 

angular velocity of, 163, 183 

mass of, 153 

radius of, 1 53 

shape of, 153, 158 

velocity of a point of, 167 
Earth's rotation, sense of, 177, 191 
Eccentricity, critical, 290 
Eddy stress, 248 
Eddy viscosity, 247 
Ekman spiral, 244 
Emagram, 35 

height computation on, 117 

precipitation lines on, 143 

Vaisala scales on, 115 
Energy, 3, 21 

conservation of, 21, 91 

dimensions of, 3 

equation of, 22, 337 

internal, 23 

kinetic, 132 

potential, 91 

specific, 3, 94 

units of, 6 
English units, 7 
Entropy, 33 

specific, 33, 46 



369 



INDEX 



Equation, Clapeyron's, 49, 50-1 

hydrostatic, 101, 188 

normal, 187, 190, 203 

of absolute motion, 154 

of continuity, 250, 252, 337 

of energy, 22, 337 

of relative equilibrium, 101, 156 

of relative motion, 171, 174, 337 
components of, 186 
projections of, 174 

of state, 12, 337 
of dry air, 14 

of mixture of perfect gases, 1 5 
of moist air, 63 
of perfect gas, 13 
of water vapor, 44 

Poisson's, 28 

tangential, 187, 188, 351 

tendency, 256 

thermal wind, 212, 214, 216, 311 

vertical, 187 
Equations, atmospheric, 337 

component, of relative motion, 186 

of horizontal flow, 186, 187, 188, 203 

perturbation, 339 

Equilibrium, equation of relative, 101, 
156 

hydrostatic, 102 

indifferent, 127, 131, 138 

mechanical, 88 

of zonal motion, 308 

phase, 42, 48, 50 

pressure in relative, 97 

relative, 101, 102, 156 

stable, 127, 130, 138 

thermal, 10 

unstable, 127, 130, 138 
Equilibrium curves, 298 
Equiscalar surface, 83, 96 
Equiscalar unit layer, 97 
Equivalent potential temperature, 76 
Equivalent temperatures, 77 
Erg, S 

Evaporation, latent heat of, 47 
Evaporation curve, 50, 51 
Evaporation temperature, 50 
Exact differential, 32, 92 
Export, 331 

Fahrenheit degree, 12 



Field, 81 

scalar, 83 

First law of thermodynamics, 2 1 
Flow, antibaric, 190, 204 

anticyclonic, 191, 204, 207 

baric, 190, 201, 204 

cyclonic, 190, 204, 207 

cyclostrophic, 199 

geostrophic, 176, 192, 233, 245, 263 

geostrophic gradient, 210, 348 

gradient, 189,265,348 

great circle, 233 

horizontal, 173 

equations of, 186, 187, 188, 203 

inertial, 190, 196, 200 

irrotational, 333 

potential, 334 

steady, 189, 208,352 

zonal, 158, 170, 306 
Flow pattern, 189 

closed, 264, 288, 294 

wave-shaped, 264, 328, 341 
Force, 3 

centrifugal, of a point of the earth, 89, 
156 

Coriolis, 172, 184 
components of, 184 
horizontal, 185 

frictional, 233, 2^8 

inertial, 155, 156, 171, 172 

of gravitation, 88, 152 

of gravity, 88, 156 

components of, 96, 185 

pressure, per unit mass, 88, 97, 154 
components of, 186 
horizontal, 186 
horizontal normal, 190, 200 
per unit volume, 99 
Frame, reference, absolute, 152 

relative, 151 
Frequency, circular, 131 
Frictional divergence, 263 
Frictional force, 233, 238 
Front, 220 

boundary conditions at, 221, 225, 340 

geostrophic, 227, 229 

pressure distribution at, 224 

shear at, 228, 231 

slope of, 223, 229, 231 

wind distribution at, 226, 231 



INDEX 



370 



Front, zonal, 227 
Frontal surface, 220 

Gas constant, specific, 13, 15 
for dry air, 14, 16 
for mixtures, 16 
for moist air, 63, 67 
for water vapor, 44 

universal, 15 

Gases, perfect, 13, 15, 24, 25 
equation of state of, 13 
laws of, 12 
mixture of, 15 
specific heats of, 25 

permanent, 11, 43 
Gauss's theorem, 333 
Generalized hydrostatic equation, 188 
Geodesic, 181 
Geodesic curvature, 181 
Geometric differential (6), 92, 126 
Geopotential, 93, 157 
Geopotential level, 92 
Geopotential unit layer, 94 
Geostrophic deviation, 234, 244 
Geostrophic flow, 176, 192, 233, 245, 263 
Geostrophic front, 227, 229 
Geostrophic gradient flow, 210, 348 
Geostrophic speed, 192, 203 
Geostrophic velocity, 210, 233-4 
Geostrophic wind level, 246 
Geostrophic wind shear, 212, 216, 231 
Gradient, 97 

horizontal pressure, 100 

pressure, 99 

temperature, 216 
Gradient flow, 189, 265, 348 

geostrophic, 210, 348 
Gradient wind diagram, 205 
Gravitation, force of, 88, 152 

Newton's law of universal, 88, 153 

potential of, 153 
Gravity, acceleration of, 88 

apparent, 227 

force of, 88, 156 

components of, 96, 185 

potential of (geopotential), 93, 157 

standard, 8, 120 

virtual, 187 
Great circle, 175, 180, 181, 182 

flow on, 233 



Hail stage, 68 
Heat, 20 

mechanical equivalent of, 22 
latent, of evaporation, 47 
of melting, 47 
of sublimation, 47 
of transformation, 46 
specific, 21 

at constant pressure, 23 
at constant volume, 23 
of dry air, 25 
of ice, 43 

of moist air, 64, 67, 139 
of water, 44 
of water vapor, 45 
Heat capacity, 21 
Heat transport, vertical, 139 
II eating, adiabatic, 68 

isobaric, 72 
Height, dynamic, 94 

computation of, 114, 115, 117 
by dry-adiabatic layer, 1 10 
by isothermal layer, 1 1 1 
on emagram, 117 
of homogeneous atmosphere, 103, 106, 

357 

Helmholtz' vorticity theorem, 324 
Hodograph, 148 
area under, 260 
of moving particle, 148 
shear, 212, 218 
Holland's theorem, 299 
Homogeneous atmosphere, 102, 105 

height of, 103, 106, 357 
Horizontal Coriolis force, 185 
Horizontal curvature, 178, 182, 265 
circle of, 180 
radius of, 180 

Horizontal divergence, 253, 324, 333 
Horizontal flow, 173 

equations of, 186, 187, 188, 203 
Horizontal (geopotential) levels, 92 
Horizontal path, 178 
Horizontal plane, 173 
Horizontal pressure force, 186 
Horizontal pressure gradient, 100 
Horizontal shear, 322 
Horizontal unit normal, 173 
Horizontal vector projection (vector 
component), 174 



371 



INDEX 



Humidity, relative, 58 

^specific, 57 

saturation, 58 
Humidity variables, 57 
Hydrostatic equation, 101 

generalized, 188 
Hydrostatic equilibrium, 102 

stability criteria for, 127, 130, 138 
Hydrostatic tables, Bjerknes, 111 

U. S. Weather Bureau, 114 

Ice, thermal properties of, 43 
Image curve, 300 
Image point, 66, 76, 133, 148 
Indifferent equilibrium, 127, 131, 138 

convectional, 138 

dry-, 131 

saturated-, 131 

Individual circulation theorem, 311 
Individual derivative, 146, 253 
Individual differential (d), 125, 146 
Individual lapse rate, 128, 139 
Inert ial circle, 198 
Inertial flow, 190, 196, 200 
Incrtial force, 155, 156, 171, 172 
Inertial path, 198, 200 
Inertial period, 198 
Inertial speed, 196, 203 
Inexact differential, 32 
Instability, 127, 130, 138 

absolute, 131 

conditional, 131 

convectional, 138 

dynamic, 286 

latent, 134 

shear, 277, 286 
Integral, circulation, 296 

line, 19, 32 

of a vector, 295, 296 
Integrating factor, 34 
Internal energy, 23 
Irrotational flow, 333 
Irrotational vector, 333 
Isentropic, 34; see also Adiabatic process 
Isobaric channel, 100, 263 

transport capacity of, 263 
Isobaric heating, 72 
Isobaric layer, standard, 111 

unit, 82, 102 
Isobaric process, 20 



Isobaric slope, 211 

Isobaric surface, 82, 102, 111, 211 

standard, 111 
Isobaric unit channel, 100 
Isobaric unit layer, 82 

dynamic thickness of, 102 
Isobaric volume ascendent, 213 
Isobars, 32, 100 

and streamlines, 189, 200, 270, 272, 
289, 348, 356, 362 

closed patterns of, 264, 288, 294 

horizontal, 100 
at front, 224 
in zonal flow, 160 

on sea level map, 82 

wave-shaped patterns of, 264, 272, 351 
Isopleths, of cyclostrophic speed, 199 

of geostrophic speed, 195 

of gradient speed, 204 

of inert ial speed, 197 

on thermodynamic diagrams, 35, 36, 

39, 61, 73, 74 
Isosteric process, 20 
Isosteric surface, 82, 102 
Isotherm advection, 220, 279, 285 
Isothermal atmosphere, 107 
Isothermal layer, height of, 111 
Isothermal slope, 214 
Isothermal surface, 82, 215, 283 
Isotherms, 32 

in stable baroclinic wave, 279 

of perfect gas, 20 

of water substance, 40, 41, 56 

Joule, 6 

Joule's experiment, 22 

Joule-Thomson effect, 24 

Kelvin, 11 

circulation theorem of, 313 
Kilojoule, 6 

Kinematic boundary condition, 225, 340 
Kinematics, 145 
Kinetic energy, 132 
Kinetic theory of gases, 236 

Lapse rate, 104, 129 
critical, 138 
dry-adiabatic, 107, 129 



INDEX 



372 



Lapse rate, individual, 128, 139 

saturation-adiabatic, 129, 139, 141 

unsaturated adiabatic, 129, 139 
Latent heat, of evaporation, 47 

of melting, 47 

of sublimation, 47 

of transformation, 46 

variation of, 47 
Latent instability, 134 
Latitude, 183, 184 

apparent, 229 

corresponding, 190 

degree of, 5, 192 
Length, dimension of, 2 

units of, 5, 7 
Level, anemometer, 241, 244 

apparent, 227 

base, 120 

geopotential, 92 

geostrophic wind, 246 

lifting condensation, 134 

of free convection, 134 

of non-divergence, 273, 280, 285, 
363 

sea, 93, 94, 228 
Level surface, 92 

apparent, 227 
Line integral, 19, 32 

of a vector, 295, 296 
Local circular vortex, 319 
Local coordinates, 174 
Local (time-) derivative, 253 
Logarithmic differentiation, 26 
Longitudinal divergence, 263, 268, 331 

Magnus* formula, 52 
Margules' formula, 232 
Mass, conservation of, 250 

dimension of, 2 

units of, 5, 7 
Mass divergence, 251 

horizontal, 256 

in baroclinic wave, 280, 285 

in barotropic wave, 274 

in closed flow pattern, 291 

in tendency equation, 257 

longitudinal, 263, 268, 331 

transversal, 264, 271 
Mass transport in surface layer, 239 
Mass variables, 9 



Maximum anticyclonic pressure field, 

162, 201 

Maximum speed (anticyclonic), 201, 204 
Maxwell's theory of viscosity, 237 
Mean free path, 237 
Mechanical equilibrium, 88 
Mechanical equivalent of heat, 22 
Melting, latent heat of, 47 
Melting point, normal, 53 
Melting pressure, 53 
Melting temperature, 53 
Meridional isobars in zonal flow, 159, 161 
Meridional plane, 157 

circulation in, 309 
Meter, 4 

dynamic, 95 
Metric ton, 5 
Millibar, 7 
Mixing length, 248 
Mixing ratio, 57 

saturation, 58 
Mixture of perfect gases, 15 
Moist air, 9, 15, 57 

equation of state of, 63 

saturated, 62 

specific gas constant of, 63 

specific heats of, 64 

thermal properties of, 57, 62 
Moisture variables, 57 
Molar volume, 14 
Mole, ton, 14 
Molecular viscosity, 237 
Molecular weight, 14 

of dry air, \ 7 

of mixture of perfect gases, 17 

of water vapor, 44 
Momentum, 3, 6 

shear of geostrophic, 231 
Momentum transport, vertical, 237, 248 

in surface layer, 240 
Motion, component equations of relative, 
186 

equation of absolute, 154 

equation of relative, 172, 174, 337 

horizontal, 173 

irrotational, 334 

projected equations of relative, 175 

turbulent, 248 

vertical, 131, 132, 136, 142, 257, 283 

zonal, 158, 170, 306 



373 



INDEX 



Mts units, 4, 6 

Natural component equations of motion, 

186 
Natural components, of acceleration, 

151, 175 
of vector, 174 

Natural coordinates, 151, 173 
Natural unit vectors, 151, 173 
Newton's formula for viscous stress, 236 
Newton's law of universal gravitation, 

88, 153 
Newton's second law of motion, 152, 155, 

172, 174 
Normal, horizontal unit, 173 

unit, 148 

Normal atmosphere, 8 
Normal boiling point, 50, 51 
Normal equation, 187, 190, 203 
Normal melting point, 53 
Normal pressure force, 190, 200 
Number of solenoids, 306 

Oscillation, stable vertical, 131 
Osculating plane, 150 
of spherical path, 175, 177, 178 

Parcel method, 125 
Partial pressure, 15, 59, 64 
Path, 147 

amplitude of, 330, 343 

and streamline, 189, 207, 266, 329, 342 

curvature of, 149 

horizontal, 178 

horizontal curvature of, 178, 266, 331, 
348 

inertial, 198, 200 

mean free, 237 

of process, 18 

vertical, 182 

wave length of, 329, 343 
Pendulum day, 198 
Perfect gases, 13, 15, 24, 25 

equation of state of, 13 

laws of, 12 

mixture of, 15 

specific heats of, 25 
Period, inertial, 198 

of stable oscillation, 132 
Permanent gases, 11, 43 



Perturbation equations, 339 
Phase, change of, 46, 49 
Phase equilibrium, 42, 48, 50 
Phases of water substance, 42, 50, 56 
Piezotropy, 338, 340 
Poisson's constants, 28 

for dry air, 28, 31 

for moist air, 65, 67 
Poisson's equation, 28 
Position vector, 145 
Potential, 96, 333 

centrifugal, 157 

geo-, 93, 157 

gravitational, 153 

of gravity (geopotential), 93, 157 

thermodynamic, 48 

velocity, 334 
Potential energy, 91 
Potential flow, 334 
Potential temperature, 31, 66 

equivalent, 76 

virtual, 66 

wet bulb, 76 
Precipitation, 68, 141 

Bergeron's theory of, 55 
Precipitation lines on emagram, 143 
Pressure, 10 

dimension of, 3 

melting, 53 

partial, 15, 59, 64 

saturation vapor, 40, 48, 57 

sea level, 122 

standard, 8 

units of, 3, 6, 7 

vapor, 40, 58, 61 
Pressure dynamic height curve, 114, 116. 

118 

Pressure force, per unit mass, 88, 97, 154 
components of, 186 
horizontal, 186 
normal, 190, 200 

per unit volume, 99 
Pressure gradient, 99 

components of, 100 

horizontal, 100 
Pressure tendency, 255 

advective, 258 

Primitive circulation theorem, 298 
Process, adiabatic, 27, 64, 67 

atmospheric, 27, 337 



INDEX 



374 



Process, cyclic, 19 

dry-adiabatic, 31 

isobaric, 20, 72 

isosteric, 20 

isothermal, 19 

path of, 18 

physical, 18 

piezotropic, 338, 340 

pseudo-adiabatic, 68, 69 

reversible saturation-adiabatic, 68, 70 

saturation-adiabatic, 72 

unsaturated adiabatic, 64 
Process curve, 108 
Process differential (d), 92, 125 
Procession, 296 
Product, scalar, 87 

scalar triple, 166 

vector, 164 

Projection of a vector, 1 74 
Propagation, speed of, 255 
Pseudo-adiabatic process, 68, 69 
Psych rometric formula, 79 

Radiation, effect on atmospheric process, 

27 

Radius, of curvature, 150 
angular, 177, 183, 289 
horizontal, 180 
vector, 151 
of earth, 153 
Rain stage, 68 
Rectangular components, of vector, 85, 

174 

of vector product, 165 
Rectangular (Cartesian) coordinates, 83 
Rectangular unit vectors, 86 
Reference frame, absolute, 152 

relative, 151 
Regelation, 54 
Relative acceleration, 170 

natural components of, 175 
Relative change in moving pressure field, 

254 

Relative circulation, 314, 317 
Relative circulation theorem, 317 
Relative equilibrium, 101, 102, 156 
equation of, 101,*156 
pressure in, 97 
Relative frame, 151 
Relative humidity, 58 



Relative motion, component equations 
of, 186 

equation of, 171, 174, 337 
Relative streamline, 277 
Relative velocity, 169 
Relative vorticity, 322 

theorem of, 324 
Relative zonal wind, 268 
Resultant of vectors, 84, 86 
Reversible saturation-adiabatic process, 

68, 70 

Right-handed screw rule, 83 
Rossby, 67 

vorticity theorem of, 324 
Rotation, 83, 151, 163, 175 

Saturated moist air, 62 
Saturated stage, 62 
Saturation, super-, 55 
Saturation-adiabatic lapse rate, 129, 139, 

141 
Saturation-adiabatic process, 72 

reversible, 68, 70 
Saturation adiabats, 72 

on diagram, 74 
Saturation mixing ratio, 58 
Saturation specific humidity, 58 
Saturation vapor pressure, 40, 48, 57 

over ice, 52 

over water, 50 
Scalar, 83 
Scalar field, 83 
Scalar product, 87 
Scalar triple product, 166 
Screw rule, right-handed, 83 
Sea level, 93, 94, 228 
Sea level altitude, 121 
Sea level pressure, 122 
Second (solar), 5 
Sense, anticyclonic, 191, 317, 323 

cyclic, 176 
of circulation, 317 
of curvature, 176, 190 
of earth's rotation, 177 
of vorticity, 323 

cyclonic, 191, 317, 323 
Shear, frontal, 228, 231 

horizontal, 322 

of geostrophic wind, 212, 216, 231 
Shear hodograph, 212, 718 ' 



375 



INDEX 



Shear instability, 277, 286 

Sidereal day, 155 

Sinusoidal flow pattern, 269, 341 

Slice method, 135 

Slope, frontal, 223, 229, 231 

isobaric, 211 

isothermal, 214 
Small circle, 175 
Snow stage, 68 
Solar day, 155 
Solar second, 5 
Solenoids, 300, 303 

in zonal motion, 307, 361 

number of, 306 

pressure-temperature, 303 

pressure-volume, 300 

temperature-entropy, 303 

theorem of, 301 
Bounding curve, 103, 108, 130 
Specific, 3 
Specific energy, 3, 94 

units of, 6, 94 
Specific entropy, 33, 46 
Specific gas constant, 13, 15 

for dry air, 14, 16 

for mixtures, 16 

for moist air, 63, 67 

for water vapor, 44 
Specific heat, 21 

at constant pressure, 23 

at constant volume, 23 

of dry air, 25 

of ice, 43 

of moist air, 64, 67, 139 

of water, 44 

of water vapor, 45 
Specific humidity, 57 

saturation, 58 
Specific volume, 3, 6, 9 
Speed, 146 

angular, 151 
of earth, 155 

critical, 268, 271, 344 

cyclostrophic, 199, 203 

geostrophic, 192, 203 

inertial, 196, 203 

maximum anticyclonic, 201, 204 

of barotropic wave, 276, 356 

of propagation, 255 

of wave, 269 



Speed, rate of change of, 150, 189 

subcritical, 268 

subgeostrophic, 207 

supercritical, 268 

supergeostrophic, 207 
Spiral, Ekman, 244 
Stability, see also Instability 

absolute, 130 

convectional, 138 

criteria in hydrostatic equilibrium, 
127, 130, 138 

region of, 219 

tendency of, 220 
Stable equilibrium, 127, 130, 138 
Stable vertical oscillation, 131 
Stable wave, 277 

baroclinic, 279 
Stage, dry, 62 

hail, 68 

rain, 68 

saturated, 62 

snow, 68 

unsaturated, 62 
Standard altitude, 119 
Standard atmosphere (U. S.), 119 
Standard component equations of rela- 
tive motion, 186 

Standard components of vector, 85, 174 
Standard coordinates, 83 
Standard gravity, 8, 120 
Standard isobaric layers, 111 
Standard isobaric surface, 111 
Standard temperature, 119 
State, 9 

critical, 42 

equation of, 12, 337 
of dry air, 14 

of mixture of perfect gases, 15 
of moist air, 63 
of perfect gas, 13 
of water vapor, 44 

triple, 41, 42 

Steady flow, 189, 208, 352 
Stokes's theorem, 320 
Stream function, 335 
Streamline, 189 

amplitude of, 269, 330, 345 

and path, 189, 207, 266, 329, 342 

horizontal curvature of, 207, 267, 289, 
322, 330 



INDEX 



376 



Streamline, relative, 277 

sinusoidal, 269, 329, 342, 348 
Streamlines and isobars, 189, 200, 270, 

272, 289, 348, 356, 362 
Stress, eddy, 248 

viscous, 235 
Stuve diagram, 30, 35 
Subcritical speed, 268 
Subgeostrophic speed, 207 
Sublimation, latent heat of, 47 
Sublimation curve, 49, 50, 53 
Sublimation temperature, 52 
Supercooling, 55 
Supercritical speed, 268 
Supergeostrophic speed, 207 
Supersaturation, 55 
Surface, equiscalar, 83, 96 

frontal, 220 

isobaric, 82, 102, 111,211 

isosteric, 82, 102 

isothermal, 82, 215, 283 

level (gcopotential), 92 

thermodynamic, 56 
Surface layer, 233 

mass transport in, 240 

(7-diagram, 49, 50 
Tangent, unit, 146, 173 
Tangential acceleration, 150, 175, 189 
Tangential equation, 187, 188, 351 
Temperature, 10, 34 

absolute, 11, 34 

dew point, 77 

dimension of, 2, 12 

equivalent, 77 

equivalent potential, 76 

evaporation, 50 

melting, 53 

potential, 31, 66 

sublimation, 52 

thermodynamic definition of, 34 

virtual, 63, 67 

virtual potential, 66 

wet bulb, 77, 78 

wet bulb potential, 76 
Temperature gradient, 216 
Temperature scales, 12 
Tendency, advective pressure, 258 

of vertical stability, 220 

pressure, 255 



Tendency equation, 256 

Tephigram, 38, 74 

Terminal curve (of vector), 147 

Thermal equilibrium, 10 

Thermal wind equation, 212, 214, 216, 

311 

Thermal wind in zonal flow, 309 
Thermodynamic definition of tempera- 
ture, 34 
Thermodynamic diagrams, 34 

(,-/>)-, 18, 29, 300 

emagram, 35 

height evaluation on, 115 

important criteria of, 35 

precipitation lines on, 143 

saturation adiabats on, 74 

Stuve, 30, 35 

(7-, 49 

tephigram, 38, 74 

vapor lines on, 61, 74 
Thermodynamic potential, 48 
Thermodynamic surface of water sub- 
stance, 56 

Thermodynamics, first law, of 21 
Time, dimension of, 2 

units of, 5, 155 
Ton (metric), 5 
Ton mole, 14 

Transformation, latent heat of, 46 
Transformation temperature, 48 
Transport, of mass in surface layer, 239 

vertical heat, 139 

vertical momentum, 237, 248 
Transport capacity of isobaric channels, 

263 

Transversal divergence, 264, 271 
Triple point, 50, 51, 53 
Triple product, scalar, 166 
Triple state, 41, 42 
Turbulence, 248 

Unit channel, isobaric, 100 
Unit layer, equiscalar, 97 

geopotential, 94 

isobaric, 82, 102 
Unit normal, 148 

horizontal, 173 
Unit tangent, 146 
Unit vectors, 85 

natural, 151, 173 



377 



INDEX 



Unit vectors, rectangular, 86 
Units, cgs, 5, 6 

English, 7 

mts, 4, 6 

Universal gas constant, 15 
Unsaturated adiabat, 65, 139 
Unsaturated-adiabatic lapse rate, 129, 

139 

Unsaturated-adiabatic process, 64 
Unsaturated air, 62 

adiabatic process of, 65 
Unsaturated stage, 62 
Unstable equilibrium, 127, 130, 138 
U. S. standard atmosphere, 119 
U. S. Weather Bureau hydrostatic 
tables, 114 

Vaisala scales on emagram, 115 
Vapor, water, equation of state of, 44 
specific gas constant of, 44 
specific heats of, 45 
Vapor lines on diagram, 61, 74 
Vapor pressure, 40, 58, 61 

saturation, 40, 48, 57 
over ice, 52 
over water, 50 
Variables, moisture, 57 

physical, 9, 32 
fields of, 82 
Vector (s), 85 

components of, 85 

differentiation of, 146, 165, 251, 312 

irrotational, 333 

line integral of, 295, 296 

natural components of, 174 

position, 145 

potential, 96, 333 

projections of, 174 

rectangular components of, 85, 174 

resultant of, 84, 86 

scalar product of, 87 

scalar triple product of, 166 

unit, 85 

vector product of, 164 
Vector area, 260, 315, 316 
Vector curvature, 151 
Vector product, 164 

components of, 165 
Vector radius of curvature, 151 
Velocity, 3, 145; see also Speed 



Velocity, absolute, 168 

angular, 3, 163 

of the earth, 163, 183 

dimensions of, 3 

geostrophic, 210, 233-4 

geostrophic deviation, 234, 244 

hodograph of, 148 

of a point of the earth, 167 

of propagation, 255 

relative, 168 

units of, 6 

Velocity potential, 334 
Vertical curvature, 182 
Vertical equation, 187 
Vertical heat transport, 139 
Vertical momentum transport, 237, 248 
Vertical motion, in cumulus convection, 
136 

in long waves, 283 

in stable oscillation, 131 

kinetic energy of, 132 
Vertical path, 182 
Vertical plane, 182 
Virtual gravity, 187 
Virtual potential temperature, 66 
Virtual temperature, 63, 67 
Viscosity, eddy, 247 

molecular, 236, 237 
Viscous stress, 235 
Volume, 2, 9 

molar, 14 

specific, 3, 6, 9 
Volume ascendent, 301 

isobaric, 213 
Vortex, circular, 308 

local, 319 
Vorticity, 320 

absolute, 322 

anticyclonic, 323 

cyclonic, 323 

in natural coordinates, 321 

in rectangular coordinates, 321 

relative, 322 

sense of, 323 

theorem of absolute, 323 

theorem of relative, 324 
Vorticity theorem, of Helmholtz, 324 

of Rossby, 324 

Water, density of, 5, 44 



INDEX 



378 



Water, thermal properties of, 44 
Water substance, critical constants for, 

42 

isotherms of, 40, 41, 56 
phases of, 42, 50, 56 
thermodynamic surface of, 56 
Water vapor, equation of state of, 44 
molecular weight of, 44 
specific heats of, 45 
Wave, baroclinic, 279, 360 

dynamic instability of, 286 
formation of, 281 
speed of, 274 
barotropic, 274, 341 
pressure field in, 347 
speed of, 276, 356 

pressure changes in, 274, 276, 281, 357 
speed of, 269 
Wave length, 269 

of streamline and path, 329, 343 
Wave number, 269 

angular, 269 
Wave-shaped flow pattern, 264, 328, 

341 

Weight, molecular, 14 
of dry air, 17 

of mixture of perfect gases, 17 
of water vapor, 44 
of atmospheric col"" 1 " 1^3,286 



Wet bulb potential temperature, 76 
Wet bulb temperature, 77, 78 
Wind, cyclostrophic, 199 

geostrophic, 192, 210 

gradient, 189, 200 

relative zonal, 268 

shear of, 212, 218,322 

thermal, 212, 309 

zonal, 160 
Wind distribution in the surface layer, 

241 

Wind shear in the surface layer, 244 
Work, 3, 5, 6, 7 

in thermodynamics, 17 

of gravity, 91 

Zonal circle, 156 

Zonal flow, 158, 170, 306 

absolute and relative, 158, 170 

acceleration in, 159, 171 

angular speed in, 160 

circulation in, 314 

equilibrium of, 308 

pressure field in, 161 

solenoids in, 307, 361 

thermal wind in, 309 
Zonal front, 227 
Zonal plane, 156 
Zonal wind, relative, 268